Lecture Notes in Physics Edited by H. Araki, Kyoto, .I. Ehlers, Menchen, K. Hepp, Z~rich R. Kippenhahn, M(Jnchen, H. A. ...
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Lecture Notes in Physics Edited by H. Araki, Kyoto, .I. Ehlers, Menchen, K. Hepp, Z~rich R. Kippenhahn, M(Jnchen, H. A. WeidenmfJIler, Heidelberg and J. Zittartz, KSIn Managing Editor: W. Beiglb6ck
251 R. Liebmann
Statistical Mechanics of Periodic Frustrated Ising Systems
Springer-Verlag Berlin Heidelberg New York Tokyo
To Gudrun Behnke, for substantial help and encouragement
S T A T I S T I C A L M E C H A N I C S OF P E R I O D I C FRUSTRATED
ISING SYSTEMS
Rainer L i e b m a n n Max-Planck-Institut Heisenbergstr.
fur F e s t k ~ r p e r f o r s c h u n g ~) I, D-7OOO S t u t t g a r t 80
CONTENTS
I.
2.
I n t r o d u c t i o n and survey
I
1.1
C r i t i c a l p h e n o m e n a at second order phase t r a n s i t i o n s
4
1.2
Scope of this book
6
One-dimensional 2.1Periodic
3.
f r u s t r a t e d Ising systems
7
ANNNI-chain
7
2.1.1
G r o u n d s t a t e d e g e n e r a c y of the A N N N I - c h a i n
2.1.2
Periodic ANNNI-chain
for
T ~ O
9
11
2.2
D e c o r a t e d chains
2.3
P a r t i a l l y f r u s t r a t e d chains
2O
2.3.1
P e r i o d i c f r u s t r a t e d chains
21
2.3.2
R a n d o m f r u s t r a t e d chain
22
Two-dimensional 3.1
15
f r u s t r a t e d Ising systems
Transformations
26
of Ising systems
26
3.1.1
Duality transformation
29
3.1.2
Decimation transformation
34
3.1.3
a)
Decoration-iteration
b)
Star-triangle
transformation
transformation
C o n n e c t i o n between d i f f e r e n t lattices
present address: AEG Aktiengesellschaft,
Sedanstr.
10, D-7900 UIm/FRG
34 35 37
VI 3.2
T r i a n g u l a r lattice
38
3.2.1
39
3.4
Simple lower bound
41
Pauling m e t h o d
42
c)
Systematic cluster a p p r o x i m a t i o n
42
P a r t i t i o n function and exact isotropic system
3.2.3
Specific heat near the frustration points (J1 = J2)
48
3.2.4
Pair correlation function, (J1 = J2)
51
GS
entropy of the 46
d i s o r d e r lines
Mapping to the q u a n t u m xy-chain and to the kinetic nn Ising chain
54
Further frustrated s y s t e m s with n o n c r o s s i n g interactions
61
3.3.1
Union-Jack
3.3.2
V i l l a i n ' s odd model and its g e n e r a l i z a t i o n s
68
lattice
61
a)
G r o u n d s t a t e s and phase diagrams
68
b)
C o r r e l a t i o n functions
70
c)
P e r i o d i c a l layered models
71
d)
C h e s s b o a r d model
74
3.3.3
Hexagon lattice
75
3.3.4
Pentagon lattice
76
3.3.5
Kagom& lattice
77
3.3.6
C o n n e c t i o n between g r o u n d s t a t e d e g e n e r a c y and existence of a phase transition at Tc = O
81
F r u s t r a t e d Ising systems with crossing interactions ANNNI-model
82
3.4.1
2d
3.4.2
Brick model
3.4.3
F r u s t r a t e d triangular lattice with n n n - i n t e r a c tions and m a g n e t i c field
88
a)
Additional nnn-interactions
89
b)
A d d i t i o n a l m a g n e t i c field
c)
Corresponding
3.4.4
4.
a) b)
3.2.2
3.2.5
3.3
E s t i m a t i o n of the g r o u n d s t a t e d e g e n e r a c y
82 87
J2 H
lattice gas model
93 97
Square lattice with competing nn- and n n n - i n t e r actions: relation to vertex m o d e l s
99
a)
System w i t h o u t m a g n e t i c field
99
b)
Systems with m a g n e t i c field
101
c)
Connection to vertex models
105
Three-dimensional
f r u s t r a t e d Ising systems
4.1
fcc
antiferromagnet
4.2
Fully and p a r t i a l l y f r u s t r a t e d simple cubic lattice
109 109 112
VII
5.
4.3
AF
pyrochlore
4.4
ANNNI-model
4.5
fcc
four-spin
model
117 122
(quartet)
model
127
Conclusion
131
References
133
I.
Introduction
The
present
and Survey
b o o k ~) r e v i e w s
systems I with
competing
configuration
of
all
Ising
interactions
some
The tive
strength,
rate
phases.
havior.
are
spins
Even i.e.
to a m u l t i t u d e
critical
For
interactions
transitions of this
can m i n i m i z e
in I s i n g
competition the e n e r g y
in the g r o u n d s t a t e remain
no of
(T = O)
in the e n e r g e t i c a l l y
exponents
certain
ratios
becomes
at
T = O
The
present
review
now
in this
fast
may
tries
the
simplest
example
system
of t h r e e
Ising
occur,
but
with
including
to s u m m a r i z e
and
on t h e i r
also
un-
order,
nonuniversal
the d e g e n e r a c y
the p a i r
a power
rela-
incommensu-
is of s e c o n d
interactions
large,
decrease
developing
depending
transition
of the
especially may
leads,
of c o m m e n s u r a t e ,
If the c o r r e s p o n d i n g
function
with
Because
(s i = ±I)
'broken',
of the
groundstates
As
of p h a s e
configuration.
competition
different
theory
simultaneously.
interactions
favorable
the
interactions.
beof the
correlation
law or e x p o n e n t i a l l y .
the r e s u l t s
obtained
up to
field.
for c o m p e t i n g
spins
at the
interactions
corners
we
consider
of a t r i a n g l e
a
(Fig.
1.1)
the H a m i l t o n i a n H
=
-
(J1s2s3 + J 2 S 3 S l
+J3SlS2 )
(1.1)
Fig. i.i The triangle formed by the three spins si = ±i with the pair interactions Jj .
w
v
¢
In the sons
case
of a n t i f e r r o m a g n e t i c
at l e a s t
relative
one
strength
interaction of the
three
interactions
is a l w a y s
for t o p o l o g i c a l
'broken'.
interactions
J1"
Depending J2
and J3
rea-
on the a dif-
~) This book is based on the habilitation thesis of the author, which has been accepted in April 1985 by the Physics Department of the University Frankfurt, Fed. Rep. Germany.
ferent groundstate (a)
J1 ~ J2
< J3
degeneracy < 0
:
J3
Ng
occurs
is w e a k e s t ,
(Fig.
therefore
N g = 2 , (as in the configuration
(b)
J1
< J2 = J3
< O
:
either
1.2): broken
ferromagnetic
J1
= J2 = J3
< O
:
J2 or J3
either
J1'
broken
for
J2 or J3
broken
\\
,
In the
isotropic
In the
are g r o u n d s t a t e s .
from
disorder
most
case
of
sition pletely
T = O
(B) +
;
(Y)
\
(6)
ions
of t h e n e a r e s t
In the
the
resolved.
per
in the
the
magnetic
3.2 w e
shall
infinite
remains
competing
theoretical problem
lattice,
But
six o f
eight
field
triangular
finite
and
Tc = 0
(nn) even
which
interactions;
treatment
interaction
whether
the
susceptibility
spin glas
coined
glasses disorder
modelled
there
as a
between exist
al-
transition
is a r e a l p h a s e
effect,
.
we use
spin
strength
for s u c h m o d e l s
nonequilibrium
of
is the t o p o l o g i c a l is u s u a l l y
H
see t h a t
of s p i n g l a s s e s 2 T o u l o u s e 3 h a s
The question,
a dynamical
finite
in the
site
with
second
neigbor
as a c u s p
'only'
In a w e a k
also
systems
lattice.
results.
up e.g. or
H = 0
in the
is e x t r e m e ,
2 . In S e c t i o n
entropy
for
frustration
on a regular
showing
for
equivalently.
no e x a c t
of
investigations
frustration
of the m a g n e t i c
spins
(x)
the d e g e n e r a c y
by a factor
context
terms
apart
(c)
the g r o u n d s t a t e
term
both
(~)
case
isotropic
lattice
the
states
is r e d u c e d
g in the
for
The four possible configurations of broken interactions (dashed lines) in the triangle with antiferromagnetic interactions.
possible N
(~)
(s) +
I
;
"t
(c~)
Fig. 1.2
;
case),
T = 0
(a) +
Ng = 6 , configurations
/
T = 0
(s)
Ng = 4 , c o n f i g u r a t i o n s
(c)
for
is s t i l l
not
trancom-
This
complication
is one of the r e a s o n s
f r u s t r a t e d systems,
where
many exact results
are known.
stable states w i t h e x t r e m e l y glas-like behavior. teresting methods
In p e r i o d i c
this p e r i o d i c
but also w e l l
nificance.
systems with competing
For example,
systems are not only in-
interactions
in m a n y m a g n e t i c
alone
lead a l r e a d y to f r u s t r a t i o n ,
tions
in l a t t i c e s
(3d)
, additional
substances (nn)
additional
behavior with decreasing
(2d)
E v e n if they are
substances,
where
(spin)
in two o p p o s i t e d i r e c t i o n s
respectively.
can m o v e on a c i r c l e
we w i l l
(spin d i m e n s i o n
t r a n s i t i o n s e.g.
from d e s c r i b i n g m a g n e t i c
dimensional periodic
Ising system,
for c o n v e n degeneracy)
are also re-
The a b s o r p t i o n of
surface
(sub-)
can be m a p p e d on a two-
if a b s o r p t i o n takes place o n l y on a d i s c r e t e
l a t t i c e of a b s o r p t i o n sites d e t e r m i n e d by the substrate.
s i = +I
s i = -I
Ising systems
systems.
of atoms on a c r y s t a l l i n e
(n = 3)
of xy-type.
substances,
l e v a n t for v e r y d i f f e r e n t p h y s i c a l
systems,
or on a sphere
(often c o n n e c t e d to e n h a n c e d g r o u n d s t a t e
find also p h a s e
monolayers
(n = 2)
e.g. be-
of the single
A l t h o u g h we only r e g a r d Ising s y s t e m s here,
ient i n t e r a c t i o n s
Weak
from lower- to
In this r e s p e c t they d i f f e r f r o m xy- and H e i s e n b e r g
the spins
interac-
or t e t r a h e d r a
significantly.
fields the m a g n e t i c m o m e n t
ions can o n l y be o r i e n t e d
interactions
temperature.
are u s e d to d e s c r i b e m a g n e t i c
cause of local c r y s t a l
correspond
to a vacancy.
in these m o n o l a y e r s from
nn
interactions become very important.
higher-dimensional
n = I).
the i n t e r a c t i o n s
as for a n t i f e r r o m a g n e t i c
i n t e r a c t i o n s m a y also lead to a c r o s s o v e r
Ising s y s t e m s
are of g r e a t sig-
. If
c o n s i s t i n g of joint t r i a n g l e s
w e a k they can r e d u c e the g r o u n d s t a t e d e g e n e r a c y
Spins
too m e t a -
l e a d i n g to spin
s u i t e d to test a p p r o x i m a t i o n
are o f t e n not l i m i t e d to n e a r e s t n e i g h b o r s
Apart
systems
for t r e a t i n g spin glasses.
Experimentally,
where
(2d)
frustrated systems
long l i f e t i m e m a y o c c u r
Therefore,
in t h e m s e l v e s ,
for the i n t e r e s t in p e r i o d i c
at least for t w o - d i m e n s i o n a l
nn
(e.g.
interactions
to an a b s o r p t i o n
For
site o c c u p i e d by an atom,
the p r o p e r d e s c r i p t i o n of p h a s e t r a n s i t i o n s from c o m m e n s u r a t e
one needs c o m p e t i n g
to i n c o m m e n s u r a t e )
apart
interactions between more
d i s t a n t sites. Substitution
alloys
AxBI_ x
c o n s i s t i n g of two types of atoms
can also be d e s c r i b e d as Ising interstitial i , and
s
1
sites. = -I
Then
to a
systems,
si = I B
atom.
A, B
if the atoms c a n n o t go on
corresponds
to an
A
a t o m at site
As a l a s t e x a m p l e
for t h e m a n y
ments,
the
we mention
the p o s i t i o n described
racy
of the p r o t o n s
originally
temperature
of a f r u s t r a t e d
adjacent After very
to the
1.1
of t h e
vertex
system,
physical
fields,
critical
properties
defined
where
to e x p e r i -
For
e.g.
instance
the
the
spin orientation
of t h e p r o t o n
for
in f e r r o e l e c t r i c
on t h e g r o u n d s t a t e
systems
energy,
If it is p o s s i b l e
systems),
low-
degenes i = ±I
between
the
two
transitions
at
reduced
T c . Defining
terminology
M(t)
~
the
of m a g n e t i c
tp
is t h e
B
close
to the p h a s e
Tc
For
tion
length
the
for
r ~ ~
critical
properties
such phase
(n = I)
of s e c o n d behavior
quantities
further
t =
Ising
which
is f i n i t e
order
M
are
IT-Tcl/T c
for
in f e r -
character-
of t h e o r d e r
parameter
, in the
c , the
the asymptotic
systems
properties
asymptotic
critical
as t h e
finds
describing
For
. Other
heat
a n d the
at a
transitions
(1.2a)
exponent
transition.
specific ~
one
review.
the m a g n e t i s a t i o n
temperature
systems
short
transitions,
(T < T c)
Here
is a s c a l a r
of t h e i r
parameter,
(e.g.
but nonanalytic
a very
phase
to
susceptibility.
an order
ized by a continuous,
systems
Transitions
. Often
the
Ising
of t h e p r e s e n t
in t h e r m o d y n a m i c
T > Tc
phase
order
Phase
change Tc
heat and
for
survey
Order
show a sudden
to i n t r o d u c e
1.1 w e g i v e
at s e c o n d
temperature
specific
T < T c , but vanishes romagnetic
at S e c o n d
by singularities
the
of f r u s t r a t e d
in S e c t i o n
1.2 b y a d e t a i l e d
critical
are a c c o m p a n i e d
M
transitions
models.
positions
of the c o n n e c t i o n
Phenomena
Many physical
free
Ising
in S e c t i o n
Critical
well
or t h e
of ice c a n b e m a p p e d
two a l l o w e d
discussion
different
followed
in ice,
systems
transitions,
02--ions.
this
summary
of I s i n g
order-disorder
by different
degeneracy
corresponds
applications
structural
show analogous
susceptibility
parameter behavior
X , the
pair correlation
exponents
behavior
the o r d e r
correla-
function
G(r)
can be defined:
C (t)
~
t -a
,
X (t)
~
t -Y
m(H)
~
H I/6
,
G(r)
~
r 2-d-n
,
~ (t) :
t
=
~ O
t -~
at
: (I.
H = O 2b)
Experiments ferent
until
developed.
on the
o n the
of
and
1.1
Normal
d
a
20(log)
~
1.1
the
lattice
have
exponents
~
002
than
stars mark
in t h i s
one
and also
the
same
Ising
for
component,
not
systems
dif-
not be
theory 4 near-
depend , but not specific
on the
exponents.
are
5
:
d = 3
q I/4 ~ I ~
5.0±0.05
system,
as m e n t i o n e d
d
(!) on the
exponents
for
could
group
exponents
critical
15.04±0.O7
results
of v e r y
ferromagnetic
Ising
6
1.250±0.002
exact
This
dimension
d = 2 and
7/4~
O.31 2 _ +O. 0.005
with
critical
lattice
Y
If f r u s t r a t i o n o c c u r s in an I s i n g more
the
ferromagnetic
the n o r m a l
Ising
exponents.
systems
a n d the
interactions,
d = 3
I/8 ~
3 0.013±0.01
In T a b l e
the
n
classes'
and renormalization
interactions
example,
I
critical
in f e r r o m a g n e t i c
square
and
'universality
hypothesis
pair
For
d = 2
Table
identical
spin dimension
type.
triangular For
(nn)
strength
lattice
have
that whole
scaling Thus
est neighbor only
shown,
substances
understood were
have
.~^+0.002 O.bJ~_O.OO1
-
d = 2 .
the o r d e r
above.
parameter
General
may have
considerations
connection
and Alexander
h a v e b e e n w o r k e d o u t e.g. b y M u k a m e l a n d K r i n s k y 6 7 a n d P i n c u s . As the d i m e n s i o n of t h e o r d e r p a r a m e t e r
in f r u s t r a t e d
systems
not only havior Until
on
a magnetic
but
, one
now we have
yields this
d
as c o m p a r e d
field
depends
can expect
a much
richer
to the n o n f r u s t r a t e d regarded (one-spin
only
general
models
review
we will
consider
for
with
four-spin
variety
structure
and
of c r i t i c a l
interactions. and multi-spin
different
almost
lattice
be-
case.
two-spin
interaction)
more
a few results
on t h e d e t a i l l e d
universality
exclusively
interactions
are
Inclusion
of
interactions classes.
In
pair-interactions, added.
1.2
Scope
of This Book
The f o l l o w i n g ing to the systems
three
lattice
simple
interactions
e.g.
contrast
to this for
as high-
Id
to solve, transfer
for
3d
2 d
systems
cluster-approximations, Carlo
for
T = O
Chapter exact tems
(in 3.1)
generacy,
is
2.1 with
the specific
After
crossing
heat,
one,
del,
frustrated
In
and in
are available,
field-
(MF-)
group
(RG)
such
approximamethods
and
4 reviews
lattice
type differ
the
disorder
decorated
and Section
3d
a large number
Ising m o d e l
behavior
systems
function
(GS) de-
of the pair
and,
cor-
on the
In Section
without
for instance
of
sys-
on the triangu-
and the m a p p i n g
are discussed.
Ising
of Ising
the g r o u n d s t a t e
line,
differ
in
3.3
crossing GS
therefore,
entrobelong
classes. the properties
Especially,
very
(AF)
2d
to the exactly
Chapter
of the A N N N I - c h a i n
d = 2
Besides
which
3.4 summarizes
interactions.
is compared
crossing
solutions.
transformations
Ising chain
are considered,
universality
as for
the a s y m p t o t i c
the r e l a t e d
of further
Section
interaction~
interactions
methods
mean
frustrated
without
the p r o p e r t i e s
general
py and the decay of the c o r r e l a t i o n
Finally
crossing
2.2 describes
extensively.
and the kinetic
to d i f f e r e n t
systems
accord-
Ising chains.
the a n t i f e r r o m a g n e t i c
function,
interactions
with
periodic
short range
yield exact
renormalization
. Section
known.
is treated
quantum-XY-
2d
expansion,
3 is by far the longest
lar lattice
a couple
T % O
frustrated
results
relation
and for
are arranged
calculations.
in Section
and
2.3 p a r t i a l l y
systems with
systems
tion,
2 deals
of the various
only a p p r o x i m a t e
and l o w - t e m p e r a t u r e
(MC)
d Ising
m a t r i x methods
Monte
Chapter
(2 to 4) of this book
dimension
(d = I to 3)
are u s u a l l y
general
chapters
solvable
systems, much
of three
the first of them, brick m o d e l
which
and belong
depending
systems the
with
ANNNI-mo-
in Section
3.4.2.
on the specific
to d i f f e r e n t
universality
classes. Chapter
5 finally
frustrated
Ising
contains systems.
a short
summary
of this
review
on periodic
2.
One-Dimensional
To e x p l a i n lar
Ising
tioned
Frustrated
the e x p r e s s i o n cluster
as the
with
field
three
sider
one-dimensional
connecting stems they
because, show
trated
systems
can o c c u r only
for
systems.
T = 0
, as
d = I
2.1
several
2.1
Periodic
the the
2.1a
triangles
nn and n n n
~sing-model)
with
-
for
relative
J1
E
o
i
i ai+1
H
=
-
real
long
short
are
ways
more
con-
of systhem,
frus-
order
(LRO)
interactions
dimension
shown
a
we n o w
to solve
range
range
critical
to d i s c u s s
i
s
-
i
that
edges.
the
of such
to f o r m
closely
a chain
now.
By r e d r a w i n g
interactions
in a l i n e a r
-
will
chain.
J1
This
J2
field,
J
= si E
be d i s c u s s e d
it and J2
linear next
are
chain
is
nearest
versions
with
later.
is:
E
i
J2
~
(2
i ~i+2
< O
it o n l y
of one h a l f - c h a i n
oi ~i+I E
common
properties
9
B
lower
have
(AF)
a magnetic
orientation
Substituting
is the
seen,
d = I
antiferromangetic
ter w i t h o u t
with
v e r s i o n of the A N N N I - m o d e l (axial 8 ; its two- a n d t h r e e - d i m e n s i o n a l
different
Hamiltonian
=
Of c o u r s e
men-
with
one-dimensional
simplicity
possibilities
interactions
one-dimensional
H
chapter
to the h i g h e r - d i m e n s i o n a l
systems
are g o i n g
it is e a s i l y
characteristic The
we
was
six,
by d i f f e r e n t
these
triangu-
ANNNI-Chain the
2.]a')
~eighbor
later.
field
In this
of the m a t h e m a t i c a l
discussed
which
treat
the
interactions
obtained
We
similarities
In Fig.
triangles,
(Fig.
many
introduction
a magnetic
occur.
systems,
in o n e - d i m e n s i o n a l
from
In Fig.
Ising
triangles.
in s p i t e
already
Without
groundstates
frustrated
in the
antiferromagnetic
example.
degenerate
Systems
frustration,
equal
simplest
Ising
; the
,
of
with
J1
does
(see Fig.
to the o t h e r
the H a m i l t o n i a n
s i si+ I
sign
determines
(2.1)
B = J1
I)
not mat-
2.1a)
one. is m a p p e d
into
' J = J2
'
i (2.2)
the
J2
J2
J2
J2
J2
J2
J2.
J2
J2
J2
(1.) J~
Frustrated ways: (a)
i.e.
the
form tion.
Periodic
ANNNI
the
chains
(a') to (c')
with
nn-
of
which
Hamiltonian
•
(= Mock
~,/
Z(k)
=
E
{o=+_1 }
(3.1)
exp (b K(b ) o.i oj)
for n o n f r u s t r a t e d systems
(for instance w h e n all
J(b)
> O) is
p r o p o r t i o n a l to the p a r t i t i o n function of an Ising system on the dual lattice with new n n - i n t e r a c t i o n s
e
-2K ~ (b)
Equation
:
(3.2a)
-2K(b)
= J~(b)/kBT
tanh K(b)
,
(3.2a)
can be rewritten in several ways:
=
tanh K~(b)
sinh 2K(b)
sinh 2K~(b)
e
K~(b)
I n t e r e s t i n g l y Eq. systems locally,
(3.2)
(3.2b)
=
I
(3.2c)
connects the interactions of the dual
i.e. this t r a n s f o r m a t i o n is a p p l i c a b l e for ar-
bitrary inhomogeneous d i s t r i b u t i o n s of f e r r o m a g n e t i c interactions,
not only in the h o m o g e n e o u s case.
The p a r t i t i o n functions of homogeneous dual systems for large 3O lattice size are linked by
31 Z(k) 2N / 2
where and
_
(cosh
N and N ~
its d u a l
are
Z*(k)
In s u c h
_
either
in p a i r s .
Ising
square
in the o r i g i n a l
lattice, same
interactions T h e n Eq.
in
(3.3)
o n e has
type
N = N*
as the o r i g i -
2d
systems),
simplifies
in
to
(3.4)
systems
connected
c a s e Eq.
2K c
sites
.
(cos 2K*) N
K = K*
sinh
the
holds.
(3.3)
Z (k*)
2K) N
self-dual
In t h i s
lattice
are of the
nn-pair
= Z(k)
Z (k) (cosh
as
interactions
(for i n s t a n c e
addition,
of
S = N + N*
is s e l f - d u a l ,
the dual
nal ones
and
,
(cosh 2K*) S/2
the n u m b e r s
system,
If a l a t t i c e If a l s o
Z* (k*)
2N * / 2
2K) S/2
=
(3.2)
singularities
b y Eq.
(3.2)
in
Z(k)
can occur
o r as a s p e c i a l
case
for
yields
I,
or transformed
Kc
This
=
(I/2)
is t h e
magnetic easily
sinh
nn
Ising
sinh
first
between
case
local
leaving
the
gauge
complex
of f r u s t r a t e d
=
of t h e
lattice.
The
result
function this
J1
are present,
or w i t h
external
one
has
frustration.
field)
to a p u r e l y
to d i s -
In the
can be transformed
ferromagnetic
system,
i n v a r i a n t 3.
is n o t p o s s i b l e ;
(3.2a,b)).
systems
can
(3.6)
without
(without
ferro-
case with different 8O a n d J2
I
interactions
(see Eq. Ising
square
anisotropic
transformations
systems
(3.5)
temperature
interactions
lattices
system
o n the
to t h e
2K2 c
the partition
In.frustrated become
transition
system
and vertical
2Klc
--~ 0 . 4 4 0 7
inverse
antiferromagnetic
tinguish
by
exact
(I+~)
be generalized
horizontal
When
in
with
As
real
the
dual
a consequence interactions
interactions the p r o p e r t i e s
on d u a l
lat-
32
t i c e s are n o t
linked
is a f u r t h e r
hint
systems
of t h e
The duality
as c l o s e l y
at the d i f f e r e n t
same
spacial
transformation
to h i g h e r - d i m e n s i o n a l where
e.g.
Refering on the
products
to t h e
as in the n o n f r u s t r a t e d
ready
that Wegner's
state
degeneracy
has
spins Ising
discussed paper
caused
been generalized
systems
2n
self-dual
fcc-lattice
of f r u s t r a t e d
This
Ising
dimension.
Ising of
behavior
case.
with multispin
occur
Wegner
32
interactions,
in the H a m i l t o n i a n .
system with
in C h a p t e r
contains
b y F.J.
four-spin
4, w e p o i n t
also
the
not by frustration,
case
interactions
out here
al-
of h i g h g r o u n d -
but by
local
gauge
symmetry. To explain two
this
systems
four-spin
local
symmetry,
(M22 a n d M32
interactions
are
in Fig.
3.2a,b
in the n o m e n c l a t u r e
v
of ref.
cells
of
32) w i t h
shown.
p •
f,
the u n i t
A
k/
A
kJ f
.
b--
,,
(a)
(b) (c)
32 (a), (b) : Two-, three-dimensional lattice gauge model of Wegner The spins (.) are in the middle of the edges of the squarerespecitve cubic lattice, whose sites are marked by crosses (x) . The four-spin interactions are drawn as hatched squares. (c) : Local symmetry: Flipping spins i to 4 for d = 2 leaves the Hamiltonian invariant, as two spins change sign in each fourspin interaction.
Fig. 3.2
F r o m Fig.
3.2c
of c o n v e n i e n t twofold
the
degeneracy
per marked
invariance
clusters
lattice
of t h e H a m i l t o n i a n
of s p i n s
of e a c h site.~
is e v i d e n t .
state,
Thus
under one
the
flipping
obtains
a
n o t o n l Y of t h e g r o u n d s t a t e ,
33
As there are two r e s p e c t i v e in the two- r e s p e c t i v e generacy
for the m o d e l s
boundary
effects)
state
entropy
is
three
spin
sites
three-dimensional M22
2 N/2
per spin
and M32
and 2 N/3
per m a r k e d
case,
with
N
and the
lattice
site
the g r o u n d s t a t e spins
de-
(without
corresponding
ground-
is
(22) So
=
I ~ in 2
(d = 2)
S(32) o
=
I ~ in 2
(d = 3)
(3.7)
and
For c o m p a r i s o n
the e n t r o p y
for
.
T ~ ~
(3.8)
for a r b i t r a r y
Ising
systems
is
S
=
in
The m o d e l tions dual
2
M32
is dual
on the simple a phase
transition
is a r e m a r k a b l e finite The
first
groundstate
Ising
systems
to the usual
cubic
of second order
example
entropy
The d u a l i t y several don't
Tc
with multispin
interactions
introduced
interest,
in q u a n t u m
e.g.
Tc ~ O
.
by Weg-
similar m o d e l s
play an i m p o r t a n t
latwith
role.
can also be g e n e r a l i z e d to spins w i t h 33 (n = 2) , but here we
for X Y - m o d e l s
this further.
of a
as they are the s i m p l e s t
chromodynamics
of f r e e d o m
transformation
components;
its
and it
in spite
with
spin degrees
discuss
like
a transition
tice g a u g e models; general
at a finite
for a system w h i c h
nn-interac-
it has
shows
ner are of such p h y s i c a l
more
Ising m o d e l w i t h
lattice 32. Therefore,
34
3.1.2
Decimation
.
.
Already chains can
.
.
.
.
.
.
.
in the
is n o t
first
.
over
.
.
.
.
.
.
.
.
.
single
.
.
.
.
.
.
of the
spins.
between
spins
the r e m a i n i n g
of
(ref.
and multi-decorated
the partition
One obtains
or g r o u p s
system
of s i m p l e
systems,
of the r e m a i n i n g
.
.
.
.
the
.
.
1 in
.
.
.
but
spins
34).
system,
function
one
a temperature
de-
spins.
This method
can be applied interact
When
they
with
only
interact
on obtains
in
with
also new
three-
Transformation .
.
.
.
simplest
with
K I and K 2
.
.
.
.
case
.
.
.
.
.
.
one obtains
(KI+K2) h
\cosh
(KI-K2) /
.
shown
two n e i g h b o r i n g
/C ° s h
in Fig. spins.
3.3a,
where
For different
the e f f e c t i v e
a cenoriginal
i n t e r a c t i o n 34
(3 9)
'
symmetrical
=
tanh K I tanh K 2
KI = K2 = K
this
direct
contribution
chain,
Eq. the
(2.25). inversion
yields
This
not
a c t on the
discuss
also
K'
transformation
frustration
= I/2
transformation
of t h i s
star-triangle
in c o s h
of the
can always because
cannot
to a m a g n e t i c
intermediate
applications
on t h e
again
is n o t u n a m b i g u o u s
decoration-iteration spins,
(3.10)
to the n n n - i n t e r a c t i o n
nnn-interactions
intermediate
section
talked
for c o m p u t i n g
interactions.
2
t a n h K'
direct
.
we have
where
part
.
interaction
interacts
-
or m o r e
We
.
to o n e - d i m e n s i o n a l
consider
spin
K'
does
.
chapter
spins
interactions
The
.
Decoration-Iteration .
ever,
.
2.1),
where
or f o u r
3.1.2a
.
of the r e m a i n i n g
four-spin
For
.
effective
two spins
tral
Transformations
restricted
all c a s e s ,
three
.
last
sum exactly
and
.
(see Fig.
pendent
We
.
2K
simple
, the
in-
decorated
be i n v e r t e d .
How-
in a c h a i n w i t h o u t
occur.
can b e g e n e r a l i z e d field
(ref.
34),
as
to s e v e r a l long
as it
spins. transformation
transformation.
in the n o w f o l l o w i n g
36
KI (a)
"--
C
K'
K2 ."
_
K~ K~ K3
(b}
(c)
y K23
Fig. 3.3
Decimation transformations: (a) Decoration-iteration transformation, (b) star-triangle transformation, (c) star-square transformation.
3.1.2b
S~a~zTEian~le Transformation
This t r a n s f o r m a t i o n is completely analog to the d e c o r a t i o n - i t e r a t i o n transformation,
only the i n t e r m e d i a t e spin or group of spins is now
i n t e r a c t i n g w i t h three other spins. Figure 3.3b again shows the simplest case. W i t h o u t a m a g n e t i c field the t r a n s f o r m a t i o n from the star i n t e r a c t i o n s (no primes) is30,34:
to the triangle interactions
(primes)
36 ~o
=
cosh
(KI+K2+K3)
~I
=
cosh
(-KI+K2+K3)
~2
=
cosh
(KI-K2+K 3)
~3
=
cosh
(KI+K2-K 3)
(3.11)
4K~ e
When
~o ~I ~2 ~3
-
all
star
4K~ '
e
interactions
~o ~2 #3 ~I
-
are
4K~ '
identical
e
-
(K i ~ K)
~o ~3 ~I ~2
, Eq.
(3.11)
be-
comes
4K ~ e
Independent romagnetic (3.12) the
cosh cosh
-
of the
sign
realizes,
corresponding
when
In a d d i t i o n the
of The
star
3.3c
spin
generates K i . Two
inverted
As
the
are n o t
only
for
the
the
triangle
K
becomes
is p o s s i b l e 30,
from
triangle
(K' < O)
Similar
transformation
on the real
Ising
to maps
Hamilton-
is p r e s e n t .
seven
with
new
transformation
four
cases
are
Therefore, and
neighboring
interactions
interactions
independent. special
fer-
imaginary.
star-triangle
star-square
new
are
for a f r u s t r a t e d
of the n e w ones
seven
interactions
transformation
interactions
interacts
original
they
that
also
real
transformation
ones,
inverse
interaction
with
in Fig.
interaction.
the n e w
no f r u s t r a t i o n
central
ones
K
however,
transformation
Hamiltonians
ians only,
where
(3.12)
(K' > O)
one
the d u a l i t y Ising
3K K
depend
is of m i n o r
This
K~ f r o m the four i , one is a f o u r - s p i n
nnn
this
is shown, spins.
on the f o u r
transformation importance.
original can be
37
3.1.3
Connection
The d e c i m a t i o n connections
Between
the D i f f e r e n t
transformations
between
of the inverse
triangles
of the t r i a n g u l a r
lattice,
in a d d i t i o n
the lattices
plication
to d u a l i t y
d e p i c t e d in Tables
star-triangle
to all triangles
Lattices
lattice
transformation
(Fig.
to the diced
3.4)
yield
3.1
leads
further
and 3.2.
Ap-
to half of the to the h e x a g o n
lattice.
,\\ 1,,"I",, ,,'?',, ,,"1",,\\ [,,
IIII
Fig. 3.4
After
Transformation between triangular and hexagon lattice.
decoration
mediate
\\\\
of all bonds
spin the s t a r - t r i a n g l e
tice.
The c o n n e c t i o n s
derived
tions
b e t w e e n the v a r i o u s 3O 35
of the h e x a g o ~
lattice w i t h
transformation from d u a l i t y
hexagon
lattices
leads
an inter-
to the K a g o m ~
and d e c i m a t i o n of Table
3.1
lat-
transforma-
are summed up
in Fig.
As to the square formation
lattices
and the Union
Jack
lattice
m o d e l 35'36 w h i c h w i l l In the f o l l o w i n g mentioned fects.
of Table
3.2,
one can show the t r i a n g u l a r
in m o r e
to be special
be further
sections detail,
using
lattice,
we discuss with
cases
discussed
the s t a r - s q u a r e Villain's
of Baxter's
in Section
regard
8-vertex
3.3.4.
the t w o - d i m e n s i o n a l
special
trans-
odd m o d e l
Ising
system~
of the f r u s t r a t i o n
ef-
38
Fig. 3.5
3.2
Connection between the hexagon lattices of Table 3.1, derived from duality (d) , star - triangle (Y-A) and decoration-iteration (I) transformations (ref. 30).
Triangular
Lattice
The a n t i f e r r o m a g n e t i c first f r u s t r a t e d appe138
using
exact
mined
the p a r t i t i o n
only appel
solution
and also
considered treated
system on the t r i a n g u l a r
has been
transfer-matrix
ger's
tities,
Ising
system,
investigated
methods
already
of the f e r r o m a g n e t i c function
and several
f o u n d the t r a n s i t i o n the case of i s o t r o p i c
the a n i s o t r o p i c
e nted n n - i n t e r a c t i o n s
case,
have d i f f e r e n t
lattice,
a few years
square related
lattice.
Tc
(h O)
nn-interactions, the three
values
(Fig.
after OnsaThey deter-
thermodynamic
temperature
where
the
by W a n n i e r 37 and Hout-
whereas
differently
3.6).
quan-
. Wannier Houtori-
39
31
A
W
Fig. 3.6
Unit cell of the anisotropic triangular lattice.
In the p a r a m e t e r ficient
space
to c o n s i d e r
simultanuous
change
ic q u a n t i t i e s only every fixed
interaction
of a c o m m o n
is f l i p p e d ;
always
triangle
ishes mark
is f e r r o m a g n e t i c Along
(here
the e d g e s
these
e.g.
o v e r the o t h e r
GS's without
chains
square
because
of the third,
(JIJ2J3),
Its c o n t o u r s
F
the
(J1,-J2,-J3),
to c o n s i d e r AF I, AF 2 and are d e f i n e d
. I n s i d e this t r i a n g l e marks
lattices.
the i s o t r o p i c interactions The p o i n t s
van-
Qi
lattices.
(full lines),
J2 = -J1 ) . The
. The p o i n t
. Also
field),
the c o r n e r s
J3 + J1 = 0
sphere,
the t h e r m o d y n a m -
it is s u f f i c i e n t
lines one of the t h r e e
to a n i s o t r o p i c square
J3 ) d o m i n a t e s
J3-direction),
(F)
the d a s h e d
corresponding the i s o t r o p i c
Along
four p o i n t s
3.7 w i t h
and
(> O)
leaves
in the d i r e c t i o n
one f o u r t h of the surface.
J1 + J2 = O , J2 + J3 = 0
ferromagnet.
factor
it is suf-
on a u n i t
of a m a g n e t i c
Therefore,
in Fig.
interactions
properties
scaling
(in the a b s e n c e
are e q u i v a l e n t .
AF 3 , c o n t a i n i n g
GS
(GS)
of s i g n of two i n t e r a c t i o n s
invariant
the s p h e r i c a l
the
of the t h r e e
s e c o n d r o w of spins o r i e n t e d
(-J1,-J2,J3) only
J1' J2' J3
the g r o u n d s t a t e
as they are i n d e p e n d e n t
Ki
W
J1
~!~!2~_2~_~h~_Q[2~5~_~£~[~2Z
3.2.1
by
A
AF I - K 3 - AF 2
c o n s i s t of i d e a l l y order between
are e x a c t l y
one i n t e r a c t i o n
two of e q u a l a b s o l u t e ordered
different
decoupled
a l so for
value
Id-chains
chains. T # O
(here
(here in
At the p o i n t s . Though
the
40
J!
A%
AF,
Fig. 3.7
Parameter space of the anisotropic triangular lattice. Special points are: F
:
(I/N)
(i,i,i)
Qi
:
e.g. ( I / H )
Ki
:
e.g.
(i,O,O)
AF i :
e.g.
(i/H)
isotropic ferromagnet
(O,i,i) id
is d e g e n e r a t e ,
per
site
vanishes
proportional
we
are mainly
interested
spherical All
three
triangle
triangle,
Before turn
fully GS
system
entropy
SO
entropy the per
systems,
to in
are
have
the
GS site,
ways
for which
the
the
same
the
degeneracy
exact
no e x a c t
absolute for
eight
the
GS
is h i g h
results
which are
of
value,
and
a single
be d i s c u s s e d
So
corners
points
states
determination
to e s t i m a t e
N ~ ~
(equivalent)
frustration
Already
as w i l l
limit
entropy
N-I/2
the
(six of
on square lattice,
frustration points, T c = O , corresponding to isotropic AF on triangular lattice.
thermodynamic
frustrated.
considering
to t h r e e
d = 3)
which
interactions is
to a h i g h finite
in t h e
F
on triangular lattice,
chain, T c = O ,
(I,I,T)
GS
Here
isotropic
F
(F)
of
SO
leads inGS
below.
by Wannier,
we
first
also
for
other
(especially
for
applicable
available
the
a finite
detail
the
elementary
this In
to y i e l d
in m o r e
are
each
GS).
of
system.
triangle
are
enough
AF i
the
41
~!~H!£_b22£~_~2H~
3.2.1a
In h i s ent
paper
tropies N
Wannier
statistical
is
per the
weight, the
a
lower
SO
which
SO
taking
lattice
number
and
of
I ]
can
be
>
~
notes
for
in
S
2
that
5
in
at
least
2
--~
+
-
third
free.
are
has
shown,
the
spin
This
differ-
with
enwhere
highest
(all
spins
immediately
0.2888
nn-corrections
÷~----÷-----+
-
every
c
with
GS
on
one
yields
0.2310
/\/\/\/\/\ +----~+-----+---/\/\/\/\/\/\ +~---+------+-----+ /\/\/\/\/\/\/\ +----- +----- +--_--+--_ \ /- -\-/+\~ -/-\-/+ -\-/-\- -/+\-/- \ /+ ~\-/- \- +/-\- /- -\-/- \+ -/- \/\/\/\/\/ -
GS
, const .
configuration
completely
of
3.8a,b,c
to
account
--
classes Fig.
N -I , N - 1 / 2
The
is
in
:
o
-~
raised
different
examples So
sites.
sublattices)
bound
>
As
site
of
one
three
into
discusses
weight.
-
_
-
-
+
-
-
(a)
(3.13)
37
.
---+-----+---
/\/\/\/\/\ +------+-----+--/\/\/\/\/\/\ ---+------+------÷~_ / \ / \ / \ / \ / \ / \ / \ ---__+ ---.+------+-----+ \ /- -\-/+\- -/-\ / \ -/-\+ /- -\ / - - - + ~ \ /+ -\-/- \- -/+\- /- -\-/- \+ -/- _ \/\/\/\/\/ -
-
-
-
+
-
-
-
-
-
+
-
-
-
(b)
x O , b u t for
(3.33) T = O
53 I
I
I
7
Fig. 3.17
5
Anisotropic triangular lattice. Continuous temperature dependence of the wavevectors
I
0
2
3
q3 = ~ - 83
(b)
ql/2 = 81
and for fixed
values of J3/Jl (with J2 = Jl ) in the disordered phase (ref. 44).
(a)
0
(a)
5
4
kBT//IJ21
8
I
i
I
I
2
"it" T 3
(b)
2
3 4 ksT//I J21
5
1.5
Fig. 3.18
8
For comparison: id ANNNIchain. Wavevector q = e versus temperature for fixed values of Jnnn/IJnnl (ref. 13).
1.0 0.5
o.5
the
correlation
G°(r) second found ty
~
r-I/2 order.
for
class.
all This
t.o
1.5
length , which However,
~ is
2.5
diverges usual
the
for
because
of
the
T = Tc
at
a phase
exponent
is
Ising
systems
nonfrustrated r-I/2
2.0
kaT/I JLI
power
law
at
q =
T = 0
I/2
power
law
transition
, contrary
to
forming
a single
we
find
will
q =
of I/4
universali-
also
for
54
several, complex For
b u t not
I > I
disorder
q3(T)
For
T = O
that
J3
(r)
G(r)
behave
=
(-I)
is in the
one
G °(r) I
GS
After
first
(3)-chains
down
to
lar b e h a v i o r lattice. again to
we
For
AF
r
in the
• r -~/2
short
there
shall
last 44
cos
of Eq.
hint
; there
(3.31),
I ~
order
to the
but
is no ql/2(T)
I
Along
in
(3)-direction,
the o t h e r
the o r i e n t a t i o n s
even
however,
for the
and a p p r o a c h
of the o r d e r e d this
distance
the f a c t o r
the w a v e v e c t o r s
two di-
(3.35)
0
(see Fig.
and
form the
perfect
one m i g h t
and
another
(3.34)
to be c o m p l e t e l y
T = 0
the
from
asymptotically
0.58835
for odd d i s t a n c e
systems,
,
already
sight
J1 = J2
has
indeed
finds
~-
again
2d
systems.
over
different
r
as was m e n t i o n e d rections
frustrated
of f r u s t r a t e d dominates
line.
and
G 03
all o t h e r
behavior
r
r-I/2
qi/2
and q3 for
only
correlated
occurs.
anisotropic
qi = ~
is true
are
Very
simi-
frustrated
vary
T ~ ~
square
continuously
, corresponding
order.
~!~_~_~h~_9~_~[=~h~!~_~_~e_~h~_~!~£~!£__~a__!~!~
3.2.5
Chain In this
section
correlation tice
and
several
The m a p p i n g on the short
we w a n t
functions,
fact, range
of
to m e n t i o n used
other 2d
that
2d
thermodynamic
transfer
matrix
of P a u l i
matrices,
noninterchangeable
Ising
classical
interactions V
. This
an i n t e r e s t i n g
by P e s c h e 1 3 6
AF
to d e t e r m i n e
triangular
lat-
systems.
systems
to
Id
quantities
can be d e r i v e d operator
method
for the
can
but usually
has
exponential
factors.
quantum
of
from always
2d a
systems
Ising
Id
systems
operator,
be e x p r e s s e d
a complicated
structure
The m a t r i x
V
is b a s e d with
the
in terms because
becomes
of
espe-
55
cially
simple
I.)
One
for t w o
succeeds
commuting 2.)
with
it s u f f i c e s
Pesche136
has
lattice
and
shown,
special
ratios
First
the
tice,
satisfying
weights and Green
w I w2
When
this
V
in the
(Hamiltonian
limit).
system
the
of
the
H
Id
form (3.36b)
respective quantum
AF
odd model
lattice,
triangular
interactions.
'free f e r m i o n
=
condition
possible
the
only
model
complete
the
is a n a l o g .
o n the
condition'
Jack
XY-chain
consider
square
for
lat-
the v e r t e x
solution
w5 w6 + w7 w8
is s a t i s f i e d ,
are
the U n i o n
the procedure
to an 8 - v e r t e x
w I .... , w 8 , making 46 1964) :
often
to t h e q u a n t u m
Here we
cases
the
so-called
which
lattice,
can be mapped
f o r the o t h e r
H
systems.
c a n be m a p p e d
+ w3 w4
, (3.36a)
to w r i t e
how
H
;
describe
Villain's
triangular
a new operator
to i n v e s t i g a t e
and which
AF
V
(-H)
hermitian
with
in f i n d i n g
it is p o s s i b l e V = exp
Then
cases:
(Hurst
(3.37)
the Hamiltonlan
] =
- nE
commutes
with
lattice,
if 47
J
Jx the
x On - J y transfer
=
I
=
tanh 2 K 3
z On-1
x z On ~n+1
matrix
V
z z ? + B On °n+1 j
(3.38)
of t h e o r i g i n a l
AF
triangular
x
Jy
B
=
,
(sinh2K1sinh2K2cosh2X 3 +cosh2K1cesh2K2sinh2K3)/cosh2
K3
(3.39) Finally
the H a m i l t o n i a n
(3.38)
c a n be r e w r i t t e n
by
the d u a l
transfor-
mation Z Z °n (~n+1
=
Z Tn
X ,
no
=
X Tn-1
X ~n
'
(3 40)
56
as the H a m i l t o n i a n
HXy
where
=
- n E
only
of the q u a n t u m x x ~n Tn+1
Jx
nn-interactions
d i r e c t i o n of t h e o r i g i n a l 36 iables :
G(r)
=
i ~ £+r-I E i=I
(-1)n < e x p
where
XY-model
have
been
c.+ c i ) > 1
,
introduced
using
the W i g n e r - J o r d a n
transformation. A t the t r a n s i t i o n corresponding other
of the o r i g i n a l
XY-chain
as a f u n c t i o n
changes,
(which c a n b e p r o v e n )
yields
largest
the
In the F e r m i o n tains gap.
eigenvalue
the t h e r m o d y n a m i c s
But
for the
(a)
J
(b)
J
(c)
J
x
x
x
+
J
+ J
= J
with
the g a p v a n i s h e s .
y
y
y
HXy ~(q)
the g r o u n d s t a t e
statement
GS
o f the t r a n s f e r
spectrum
three
This
t h a t the
of the o r i g i n a l
representation
the e i g e n v a l u e
system
is t w o e i g e n s t a t e s
of t h e p a r a m e t e r s .
assumption
mines
Ising
that
wave
cross
contains
function
matrix
of the each the
of
and thus
Hxy deter-
system.
c a n be d i a g o n a l i z e d usually
having
a n d o n e ob-
a finite
energy
Cases:
=
- B
;
~ (O)
=
O
=
+ B
;
~(~)
=
O
~(e)
=
O
; IBI < 2 J -x 8 = a r c cos
;
(-B/2J x)
,
(3.43)
57
For
J1
line
= J2
I
0 in F i g . 3 . 1 6 , w h e r e a s c point X = I , T = 0 . For I
O ) - L R O
phases.
paramagnetic agreement tion
point
F and
(P)
with
along
(I =
the
T = O)
aFp
are
down decay
to of
= -
I/~
connected
transition
extends
exponential
disorder
I,
groundstates
Between phase
the
the
AF
lines the
the
to TCF
. For s > sFP and corresponding and
frustration diagonal
TCA F
the
point,
in
correlation
l i n e 13
-4K~ cosh
for
T
4K~
> O
=
e
: ( ~ r h - I /2 \2 )
G(r)
(3.50)
r (tanh
K 2)
for
even
r (3.51)
=
0
for
odd
r
func-
64
For
TD I)
-
1+~ 21 in (\--~----) ~
It is i n t e r e s t i n g an e n t r o p y way,
the
effect.
to n o t e When
intermediate
the c o u p l i n g
the J
0.2406
spins
chain
(3.60)
of n e i g h b o r i n g
on b o t h
has
chains
a finite
GS
J'
chains
are o r i e n t e d entropy
per
the
via same
site.
69
However, tropy
for o p p o s i t e
of t h e
The
PUD
tion
point
given
spin
intermediate
model
has
orientation chain
two phase
(Fig.
3.26).
The
2K s i n h
(K+K')
=
on t h e
J'
chains
the
GS
en-
vanishes.
boundaries,
both
two boundaries
ending
of t h e
at the
frustra-
F and AF
phase
are
by 59
sinh
I
and
(3.61) sinh
Like in there
2K s i n h
(K+K')
the U n i o n
Jack
exists
sublattice
LRO
=
lattice
on o n e
causes
the
- I
also
in the
sublattice,
entropy
PUD
while
to r e m a i n
model
for
the disorder
finite
down
to
I > I
on the other T = 0
.
T/J
PARA
~/
-
Lt
Fig. 3.26
The
GS
ordered
Phase diagram of the
of t h e for
ZZD
i < 1
For
I = I
del,
(I = I, T = O)
For J'
i > I
like
of s p i n s pairs
model too,
the
S
model
markedly
do n o t
interacting
form a fully
(~ick-zack
with
PUD
model (ref. 59).
o
= 0
domino),
via
form J'
frustrated
see T a b l e
3.2d,
is
F
.
it is e q u i v a l e n t
is its f r u s t r a t i o n
it d i f f e r s
interactions
PUD
from the
contiguous
point PUD
chains.
are therefore triangular
to V i l l i a n ' s
odd mo-
too. model, In t h e
coupled
lattice,
as t h e GS
rigidly.
the
GS
strong the pairs As
these
entropy
70
per pair site
is i d e n t i c a l
thus
So(~>1) The
ZZD
I sAFA 2 o
=
model
of t h e
AF
triangular
lattice;
O.1615
possesses
2 tanh
(K+K')
I > I
there
For
to t h a t
So
per
is
tanh
only
2K
=
(3.62)
an I s i n g
system remains 59 diagram is s h o w n
PARA
I < I
at 59
(3.63)
in the f r e e e n e r g y
in t h e p a r a m a g n e t i c in Fig.
for
I
is no s i n g u l a r i t y
the
transition
phase.
The
for
T ~ 0
corresponding
,
phase
3.27.
FERRO-
-
--I =J'lJ
Fig. 3.27
3.3.2b
Correlation .
Consider For
Phase diagram of the
.
.
.
.
GO(r)
.
.
.
.
.
.
PUD
and all of
chain):
.
(without
model (ref. 59).
Functions .
.
.
.
.
ZZD
.
functions models
F o r g a c s 35,
found
q = I/2
G(r)
zero and finite
sults
.
and
(odd m o d e l )
the b e h a v i o r for
.
the c o r r e l a t i o n
I < I
= I
.
ZZD
:
ferromagnetic;
. Wolff
in h o r i z o n t a l ,
signs
T = O
G°(r)
G a b a y 61 a n d P e s c h e 1 3 0
temperature.
changing
are
for
and
have
For
determined
Z i t t a r t z 54 d i s c u s s
diagonal
and vertical
Here we only mention
depending
= I
on the e x a c t
the
in d e t a i l direction
T = O
position
re-
of the
71
const.
• r-~/2
;
r
even
GO(r)
(3.64a) const./~
for horizontal
OOr>
.
For
i > I
for
T = O
functions
-
I 5 (-I)
I > I
and
>
chains
chains the
J'
of t h e
cause
PUD
model
the f i n i t e
and J
chains
GS
are
are
ordered,
entropy.
((b)
The
is o n l y
,
(3.65a)
(3.65b)
apart
and diagonal
f r o m the
entropy
direction
also
G°(r)
is e q u i v a l e n t
of t h e
to t h e
AF
ZZD tri-
result:
o
GZZD(r)
3.3.2c
~
r
-~/=
cos
{2~ ) \~- r
(3.66)
~~!_~[2[~_~2~!~
PUD
tically
and the
ZZD
models
(or h o r i z o n t a l l y )
arranged
translation
period
is
papers 54'62
have
lated
phase
the
)r
T = O
in v e r t i c a l
tition
J'
J
along
o (r) GpUD
The
and
r ~ ~):
(-I
angular
the
intermediate
=
For
directions,
odd
00
o' GpuD(r)
model
r
direction. and
the
correlation valid
and vertical
;
cos
for diagonal
whereas
• r -I/2
their
results
invariant
~ = 2
can be c o n s i d e r e d
layered
within
. Wolff,
investigated diagrams
are summarized
and
models, each
Hoever
general
and
54.
layer
as the the
simplest
interactions
and the
Zittartz
layered models
correlation
in ref.
where
functions.
layer
are
repe-
in a s e r i e s and have
ver-
of
calcu-
The method
and
72
We w a n t
to d i s c u s s
ization
of the
strating
how
little
the t r a n s i t i o n , model ones (Fig.
J1
if
model GS
3.28)
J2
where
changes
of t h e s e with
, for
models.
are related
. Calling
the
J
the
J'
first
one,
one
obtains
only
along
the t h i c k
along
these
lines
lines
and a l s o
%% %% -I %%.
I T
"%%
demon-
to the p r o p e r t i e s
parallel a
a general-
clearly
interactions
interactions
T = 0
I
The
v = 2 , is an e x a m p l e
properties
Tc > O
and m o d i f y i n g
to b e c o m e
guration
two m o r e
PUD
'phase
to the
at PUD
J1
d i a g r a m '63
So > O along
of the
The
spin
the d a s h e d
confi-
part
of
Yl
%k
--1 "I(* %
Fig. 3.28
Groundstate phase diagram of the modified
the d i a g o n a l Fig.
3.29
Y2
But o n l y
9ig. 3.29
the
Yl
+ Y2 = O
GS along
energy
in Fig. Eo
the w h o l e
3.28,
where
is p l o t t e d diagonal
PUD
Yl
model (ref. 63).
Yi = J i / J
as a f u n c t i o n + Y2 = O
The corresponding groundstate energy (ref. 63).
" In
of
one has
Yl
and Tc = O
,
73
everywhere pending
ohly
Fig. 3.30
The
else on
T = O
not have
is f o r G(r)
Ghl (r)
whereas
=
along
Gv(r)
the
and
frequent
,
< ~ r>
r
for
r
occurrence
again
ample
the
de-
c
on
(ref. 63).
o f the Tc
correlation Yl
we
chessboard
=
GS
spin
confi-
It is i n t e r e s t i n g function
< I , only
(-I) r
similar 58
for
the h o r i z o n t a l
model
such
chains
n = I/2
to t h e a n i s o t r o p i c
frus-
are
have
become
to d i s c u s s
different
also
completely
de-
.
q = I/2
systems
now going with
(3.67)
(3.68)
u p to n o w w h i c h
are
,
;
of t h e v a l u e
discussed
. However,
r ~
• r -I/2
a s s u m p t i o n 35 t h a t a l l
q = I/2
direction
the h o r i z o n t a l
for even
systems
changes
(if
Gh2 (r)
lattice
cos
for odd distance coupled,
> I
T
direction
the vertical
=
Tc
influence
and vertical
= - Y2
at a f i n i t e
are exchanged).
I
triangular
trated
occurs
3.30).
indicating
any clear
Yl
the h o r i z o n t a l
trated
transition
(see Fig.
the h o r i z o n t a l
, that
and vertical
The
ly I + y 2 1
'phase diagram' does
to c o n s i d e r
Along
Ising
The corresponding transition temperature
gurations
Tc = 0
a normal
in t h e
examples
no finite critical
Tc
of f r u s -
had
at
Tc = O
as a f i r s t
counter
behavior.
led to with ex-
74
3.3.2d The
Chessboard
chessboard
= 4
shown
frustrated. relation T =O
Model
model 59'54'64
in T a b l e
3.2e where
We mention
length
down
every
this m o d e l
to
T = O
layered model
second
as the
period
elementary
square
one with
a finite
first
, and thus
with
not becoming
critical
is corat
.
Already
Andr&
e t al.
59
and also the existence
tartz
64 h a v e
S
~
o
In ref.
As the
reason
show
rives
criteria.
T = O
GS
systems
r -q
short turn
T = O
the
entropy
So
for
. Wolff
T > 0 and
Zit-
short
correlation
function
correlation
length
by flipping with
decay,
range
which
GS
different
6o
between , that
local!y
whereas
and
is
only GS
systems
other models
demonstrate
from the previous
'superfrustration',
isolated
correlations
to t h r e e
decays
ex-
:
(3.70)
'isolated'
conjectures
for
with
He d i s t i n g u i s h e s
from other
case.
GS
the diagonal
, S ~ t o 65 c o n s i d e r s
without
thus
that
for t h i s b e h a v i o r
Tc = O
We now
of a f i n i t e
of a t r a n s i t i o n
in (I + ~ )
with
only
absence
(3.69)
for
=
spective
the
obtained:
54 t h e y
~I
found
O.371
ponentially
and
is a d i a g o n a l
frustrated GS
a limited
without
e -~/r
systems
, which
to h a v e
systems
for w h i c h
range
isolated
with
cannot
number
long
GS
he dere-
be r e a c h e d
of
spins.
should have
decay.
also exhibiting
the chessboard
model
He
correlations
exponential
decay
to b e n o p e c u l i a r
75
3.3.3 The
~9~__~!~2
hexagon
diagonal
lattice
layered
interaction
(Tab.
square
is o m i t t e d
3.1b)
lattice as
shown
can
be
where
This
way
thus
also
with
the
in e v e r y
in Fig.
/\ k.,,,.,~",~ ~..'~,. /-,,,/\/ Fig. 3.31
regarded
as
a special
second
row
every
and
the
fully
o•" K
i~
~
configuration
of
solved
Ising
interactions
the
system K. = l
general
on
the
"
± K
The dotted
anisotropic
hexagon shown
and
lattice
in Fig.
3.32.
/\/k/\ I I I \/\/\/\.
i- , ~ / ~i / \ / /J Fig. 3.32
The
Configuration of interactions of the fully frustrated hexagon lattice. Thick lines are AF , thin lines F interactions (ref. 66).
system
has
SO
O.214
~-
a finite
GS
,
entropy
66
(3.71)
a
second
\/\/
Z i t t a r t z 66 h a v e frustrated
of
3.31.
Transformation of a square lattice into a hexagon lattice. interactions have to be omitted (ref. 66).
Wolff
case
76
and
is p a r a m a g n e t i c
relation
length
-1 go
=
in
chessboard
become
critical
3.3.4
~H2D_~i~
investigated
of a l a y e r e d
down
to
T = 0
, where
the c o r -
f i n i t e 66
(3.72)
model
at
ferromagnetic
been
all t e m p e r a t u r e s
(2 + V ~ )
As t h e
The
for
remains
the
T = 0
and the
fully
hexagon
lattice
does
not
.
fully
by Waldor,
system which
frustrated
frustrated
Wolff
is s h o w n
AF
pentagon
a n d Z i t t a r t z 67, in Fig.
lattice
as a n o t h e r
have
example
3.33.
S Fig. 3.33
Pentagon lattice; one layer is drawn with thick lines (ref. 67).
Whereas
in t h e
in the
AF
S
o
~
F
case
case the
does
down
correlation
of t h e a s y m p t o t i c
also
the u s u a l
a finite
GS
Ising
behavior 67
systems
as e x p e c t e d ,
entropy
(3.73)
a n d is p a r a m a g n e t i c
the
found has
0.2336
horizontal
Like
they
system
to
T = O
length
oscillations of the
not become
. The
~h(T) of
Gh(r)
two previous
critical
at
temperature
dependence
and of t h e w a v e v e c t o r are
sections
T = O
.
shown the
in Fig. AF
of the
q = e(T) 3.34.
pentagon
lattice
77
I
2
Fig. 3.34
3.3.5
&
6
8
T/J
Horizontal correlation length {h and wavevector lattice as a function of temperature.
@
of the
AF
pentagon
~2~9_~9~9
As for the t r i a n g u l a r gom6
lattice
tions
(Table
case with
all i n t e r a c t i o n s
fore,
o
the p r o p e r t i e s for d i f f e r e n t
of the Ka-
configura-
Then
is the i s o t r o p i c
all e l e m e n t a r y
but not the hexagons.
case,
in this range.
where
in the
Kano and Naya 68 have
in the free e n e r g y
is p a r a m a g n e t i c
AF
triangles
for
T > 0
The
GS
lat-
found
; there-
entropy
is
and very high:
~
O.5018
(3.74)
A very
similar
tioned
in c o n n e c t i o n
triangles
Sp p
frustration
of a s i n g u l a r i t y
the s y s t e m
finite
lattice studied
Ki .
are equal.
are frustrated,
the absence
S
have been
of the i n t e r a c t i o n s
The s i m p l e s t
tice
and the square
3.1c)
=
value
as being
is o b t a i n e d with
from the P a u l i n g
the t r i a n g u l a r
lattice,
approximation which
treats
menthe
independent:
2 3 in 2 + ~ in ~
~
O.5014
(3 75)
78
The
reason
relation Figure
for
3.35
respective lattice
this
which
we
shows AF
good
shall the
agreement discuss
internal
interactions
r e s u l t s 68
which
#
is p r o b a b l y
energy
of t h e K a g o m &
together
for
the very weak
pair
cor-
further.
T > 0
with look
lattice
with
the corresponding
F
triangular
similar.
-0.4-O.B- 1.2-
-1.6-
-2.0-2.4]
I
-2.8
/
Kano
anisotropic
J1
three
and Naya
Kagom~
= J2 = J > O
magnetic
The
s i g n of
P
s
L'
12-,.I/ILl
1~
calculated
the partition
(with t h r e e
[J[ < 0
J
different
only
function
of the
interactions
Ki
for
G e i l i k m a n 69 for t h e c a s e
shown
in Fig.
functions
3.36a has
along
discussed
the dashed
is u n i m p o r t a n t
in the a b s e n c e
a normal
transition
of
lines a
field. J3
F
and the transition
phase, I
C
and the correlation
For weak
K c-
I
~,
nn-directions),
, J3 = - ~
diagram
3.36a.
have
lattice
different
the phase in Fig.
~-
Temperature dependence of the internal energy of the AF (upper) and F (lower) Kagom& lattice (full lines). The corresponding triangular lattice results are shown as dashed lines (ref. 68).
Although
the
I
"~l
K/011
Fig. 3.35
/
I I I I
_
(l < I)
T J
c
he f i n d s
I In 2
(1-~)
temperature
Ising for
i ~ I
to a s i m p l e
vanishes
linear:
(3.76)
79
7 f\\ ~/ × X \,
£
X - //"
(a)
Fig. 3.36
The
(b)
Frustrated Kagom@ lattice; the double lines correspond to tions J3 "
Only the triangles are frustrated, along the dashed lines the pair correlation function is discussed in ref. 69;
(b)
here the hexagons are frustrated too.
correlation
function
G(r)
completely along 69 for I ~ I : TD J
KDI
2 In 2
-
(I : I, T = O)
triangular
lattice
i > I
the
for For has
three
i > I
the
system
inves£igated
free
models
for
apart
order
and
from
TD
lines
in Fig.
(T D > T c)
3.36a
which
frustration
point
for
but no order
T ~ 0
(according
correlation
of
the pair
to ref.
functions
is p a r a m a g n e t i c
a whole
configurations energy
dashed
is a l s o
(3.77)
is the
chains
all
the line
like
for the
AF
3~16).
direction
T = 0
different the
(Fig.
J3
the p e r p e n d i c u l a r
along
a disorder
(I-I)
The point
For
interac-
(a)
vanishes linear
AF
family
for
finite
function
an a r b i t r a r y
occurs
T ~ O
in
~t ~ O
temperature. Kagom~
S H t o 70
models
with
the b e h a v i o r
of
for
T > O
. In t h e s e
part
of the h e x a g o n s
is f r u s t r a t e d . To the Fig.
right
3°36b
side
shows
of Fig.
3.36a,
one possible
where
no hexagon
realization
;
vanish).
interactions
correlation also
Gi(r)
of f r u s t r a t e d
F and AF
all t r i a n g l e s
69 for
is f r u s t r a t e d ,
of t h e o t h e r
extreme
where
80
all
hexagons
tration
the
the
complexe
the
whole
in
are
of
the
tanh
energy
corresponding
~max-1
In
the
N>
proof
function
with
w
fast;
the
=
=
agrees
=
already
in F i g .
This
means
temperature
independent
outside
the
3.37
there
is can
including
T > 0
(0.74) li-jl
,
a maximal
correlation
the
be
no
the
frus-
area
analytic
T = 0
upper
of
hatched
of
including
singularity
. For
the
cor-
b o u n d 70
(3.78)
length
0.30
(3.79)
expands of
two
<s.s > in p o w e r s o f t h e 1 3 spins on a single frustrated
w I + w + w2
KI
simplest
2
that
for
to
Itanh
-
any
shows energy
- plane
obtains
4 •
he
nn-correlation
U
at he
S~to free
axes.
J
the
(B
positive
free
relation
frustrated.
hexagons
and
=
1 - x 3 + x
x = e
approximation
< --
-21KI
1 ~
the
correlation
triangle
'
This
for
nn-pair
power
internal
(3.80)
series
converges
energy
(that
is,
very the
function)
I~nn
I
(3.81)
to w i t h i n
a few
percent
with
the
exact
U
.
4i
-4
Fig. 3.37
2
In the non-hatched area of the complexe tanh (~ IJI) - plane the free energy of all KagomA models with fully frustrated triangles is analytic.
81
Equation also
(3.78)
for
means
T = 0
tem besides lattices,
. Thus
internal
the pure
AF
(3.81)
is a good
depend
only w e a k l y
We also note
close
energy
Kagom~
U
at
integrating for
S
also
in this
U(T)
function
is the forth
pentagon T = 0
no hexagons
on the f r u s t r a t i o n
to the exact
lattice
and the e n t r o p y
approximation
that
Kagom~
critical
lattice w h e r e
approximation
of the c o r r e l a t i o n
and the f r u s t r a t e d
does not b e c o m e
The exact
decay
the f r u s t r a t e d
the c h e s s b o a r d
which
the Pauling
exponential
sys-
and h e x a g o n
. are known only
are
case,
frustrated. U(T)
for
As Eq.
and S(T)
can
of the hexagons. from Eq.
S O , Eq.
(3.81)
exactly
(3.75), w h i c h
yields
also was very
result.
~ ~ _ ~ _ _ ~ _ _ ~ ~ _ ~ _ ~ ~ _ ~ ! _ ~ _ ~
3.3.6
Transition
at
TG_[_ ~
At the end of Sections trated
Ising
systems
considerations state
3.2 and 3.3 w h e r e we have d i s c u s s e d
solved exactly,
on the c o n n e c t i o n
and the e x i s t e n c e
Hoever,
Wolff
between
If the set of all
2d
more
the d e g e n e r a c y
GS
formulated
frus-
general
of the ground-
is connected,
the f o l l o w i n g
that
the global
symmetry
, cannot be broken.
conjecture
is if any two
into each other by a series
transformations, s I ~ - sl
to m e n t i o n
of a transition.
and Zittartz 71 have
be t r a n s f o r m e d
we w a n t
of p u r e l y
GS
can
local
of the Hamiltonian,
In this
case there
is no phase
transition.
In case of the c h e s s b o a r d 71 and the connected 2 nn
by l-spin
flip processes;
spins m u s t be flipped
another
one.
For all three
systems
do not become
AF
Kagom~
simultaneously systems
critical
at
lattice
in the h e x a g o n
~(T=O) T = O
to obtain one remains
all
lattice
GS
GS
from
finite,
these
in a g r e e m e n t
with
the above
conjecture. If the Hoever
GS
are
always
are not connected, there are no g e n e r a l statements; 71 et al. m e n t i o n examples w i t h and w i t h o u t a transition.
82
S~to has tems
also put
foreward
a conjecture
consistent
with
the t h r e e
sys-
just mentioned:
If a n d o n l y
if a f r u s t r a t e d
no transition
and
~(T=O)
Ising
system
remains
also
finite,
for
the
T = O
has
set of all
GS
is c o n n e c t e d .
S~to
calls
such models
'superfrustrated'.
Both
conjectures
still
have
to b e p r o v e n .
3.4
Frustrated
In this the
section
square
brick model
2d 2d
Sec.
2.1)
GS
point The
and crossing
a n d the
nnn
we also
triangular
interactions
and finally
consider
comment
analog
disorder
and
which,
the
on the
(simplified)
connection
of
Tc $ O
the
and mean
field
Id
++++
the p h a s e
ANNNI-chain direction
(compare
and
introducing
case: =
F
for
for
I = - J2/J1
i > 0.5
< 0.5
; at t h e
frustration
occurs. to t h e
Id
. Of the w e l l
low t e m p e r a t u r e approximation diagram
c a l c u l a t i o n s 12'72
Id
J > 0 b e t w e e n s p i n s on n e i g h b o r i n g o l a t t i c e is s h o w n in T a b l e 3.2i.
to the
difference
with
f r o m the
in o n e p e r p e n d i c u l a r
, and periodic
essential
sitions
ANNNI-model
exactly.
is d e r i v e d
corresponding
are
Id
Interactions
models.
nn-interactions
diagrams
Thus
exactly
to v e r t e x
by repetition
the
(J1 > O)
2d
the A N N N I - m o d e l
ANNNI-model
chains;
nn
Crossing
ANNNI-Model
additional
The
with
With
the
be solved
solved
these models
The
we discuss
cannot
comparison
3.4.1
Systems
lattice with
therefore, For
Ising
too
(does not
(too l a r g e 3.38 low
is the e x i s t e n c e
known methods
series
of Fig.
for not
chain
to d e t e r m i n e
converge
fluctuations)
is c o m b i n e d T
of t r a n -
for cannot
from Monte
phase
d = 2) b e used.
Carlo
, the M~ller-Hartmann/Zittartz
(MC)
83
Poramognetic
2.0
1.0 Antiphase
I
I
0.2
Fig. 3.38
for d o m a i n
a p p r o x i m a t i o n 74 g o o d Adjacent
to the
finite
therefore, rate
especially
point, With
M
location
phase
1.0
Jl = (l-e) Jo
and
for
with
X < 0.5
a free
the
same
respective
order.
Not
is the o c c u r e n c e
between
and
fermion
IJ21
interesting
of the L i f s h i t z
the
point
L'
common
and,
of an i n c o m m e n s u -
and the 75
X > 0.5
P
, a special
phase;
the
multicritical
is not y e t known.
the m e t h o d
of M H l l e r - H a r t m a n n
can be e s t i m a t e d
F
phases
0.8
energies 12'73'76
T O)
now there
are many
frustration The
free
described
Figure
only
T = 0 3.39
The
shows
M
data,
configurations with
each other,
one
over
such domain
.
1 .
.
.
.
these
analytically
largest
eigenvalue
and thus
determines
visible
to the
phase.
They
in t h e
occur,
start
M
which
a minimal
distance
but
have
don't
from
phase
at
c a n be r = 2 .
to be s t r a i g h t
.
234 .
.
.
.
.
.
.
.
wall
56
.
.
.
.
.
configuration.
.
7890 .
in Fig. phase
'dislocation
and
leads
to a f r e e
is c o n n e c t e d
3.40,
q
q
increases For
wall
fermion
to t h e a v e r a g e
the w a v e v e c t o r
boundary.
free'
FFA
(ref. 74).
configurations problem
distance
where between
can the walls
. continuously
the t r a n s i t i o n s
from the
Villain
F
to
a n d B a k 74 ob-
tain -2K F
:
K I + 2K 2
=
e
o
(3.84a)
and -2K
:
at
a n d B a k 74 is v e r y
phase.
that
down
F and M
A typical domain wall configuration included in
summation
the
of the
walls
to r e a c h
the
of V i l l a i n
MC
spin
phase
occuring
.
by d o n e
As
P
all a r e
between
(FFA)
by the
do n o t t o u c h
.
Fig. 3.39
for the
the b e h a v i o r
such
phase
T = O)
b y a set of d o m a i n
the w a l l s
as for
in e q u i l i b r i u m ,
approximation
supported
low t e m p e r a t u r e
point
, M and P
(i = 0.5,
to u n d e r s t a n d
the a s s u m p t i o n
a s s u m p t i o n 12 of a L i f s h i t z F
indications
point
fermion
helpful
Thus
where
K I + 2K 2
=
- 2 e
o
;
(3.84b)
85
i 112i
I -exp (-2~ Jo)
I
0
X
2exp (-2~3 J0)
Fig. 3.40
Wavevector q of the modulated phase as a function of x = - (JI+2J2)/T (ref. 74).
the f i r s t
one
corresponds
second
one differs
(K ° >>
I)
The
FFA
rection
result
n
-~
r -~ cos
I ~
factor
for
t w o on t h e
K ° >> right
I , whereas hand
site
the
from the
(3.83).
for t h e p a i r
depending
=
(2.82)
of Eq.
of the c o m p e t i n g
G(r) with
b y the
expansion
to Eq.
correlation function 74 is :
G(r)
in the di-
interactions
~ q x
(3.85a)
continuously
on t e m p e r a t u r e
and
~ :
(l-q) 2
(3.85b)
Within
the f r a m e w o r k
walls,
the phase
o f the
boundary
FFA
between
which
assumes
M and P
nontouching
phase
cannot
domain
be
investi-
gated. The
inclusion
fects
where
in t h e also
2d
XY
found
G(r)
of a l o w c o n c e n t r a t i o n
walls
touch)
ferromagnet,
a modulated
=
where
n(T)
phases
in t h e
corresponds
of d i s l o c a t i o n s to t a k i n g
for w h i c h
care
Kosterlitz
(that is of deof t h e v e r t i c e s
a n d T h o u l e s s 77
have
phase with
r -n
(3.86)
is t e m p e r a t u r e two models
dependent.
together
with
The
equivalence
the k n o w n
Tc
of the
M
of the X Y - m o d e l
86
yields
the t r a n s i t i o n
A NNNI- m o d e 1 7 4 .
temperature
between
the
M and P
phases
of the
For ]
q
the
-i/4
; (b) :
The abscissa in
Whereas of
Wada
and
divergencies
size L ~ ~
analysis , the
determined Landau
(for exponent
from
further
sublattice
the
when
the finds
I :-I)
position that
approaching
this
must of
the
this
system,
that
thus
.
have different scale
interprete
infinite
s
i TI :
magnetisation;
phase
field
triangular
diagram
four phases
1 x 3 between
phase the
H
for
increases
the
lattice.
of K a b u r a g i H = O
occurs. ferro
further
also
and Kanamori a
On t h e o t h e r
and paramagnetic
2 x 2 hand
92
and
for
phases
dis-
94
lJ~h
oxol-I
i
Fig.
3.47
GS phase diagram of the triangular lattice with nn- and nnn-interactions Jl and J2 in a magnetic field H (ref. 92).
Consider with For
the
spins this
sition For
by
Monte
phase
Alexander to
(J2 = O)
3.48a,b.
For
I < 0
already order
been up
order.
at
Figure
value y =
is 1.42
values:
phase
3.46
H = 2.43
> O)
the
with
an
= 0.40).
, v = 0.87 ~ =
I/9
, T =
Finite
size
n = 0.27 ~
1.444
predicted
the
q = 3
finite, phase
by
the
< H
are of
~ 0.44
the (the
for
specific 'best' yields:
, very
to
close
and
M~I-
shown
in
the
~ 0.867
I
is
second
it
is
i = by
iJ11
X = -
< Hc
scaling 40-42
, ~ = 5/6
94 are
case
marked
tranmodel.
determined
transition
Ht
L a n d a u 88
points
= 6
curves
and
the
Potts
< H < Hcl
(H)
exist
antiparallel.
c renormalization
, for
divergence s/~
13/9
the
H = Ht
exponent
and
is
GS
one
have
0 T
three
on
qualitatively
H > O
determined
of
for
. The
Tc
tricritical
and
al. 86
field
agree
only
H
class
c real space
H = 0
the
. Then to
et
T
88,
For
demonstrates
s/v
finite
point
diagram
where
J1
magnetic
and
discussed.
3.49
Domany
m e t h o d 95
a tricritical
The
in F i g .
and
calculations
(J2
H > O
parallel
universality
the
ler-Hartmann/Zittartz Fig.
93
the
a transition Carlo
for
sublattices
belong
~ = 0 to
x ~
two
case
to
leads
~
on
I
has
first is
shown
crosses. heat Potts
at model
B = O.11 'exact' and
,
Potts
~ ~ 0.266
.
85
1.6
[
,
,
~
J
i
1.4 kT/J
1.2 1.0
-- o.a 0.6
0.4
0.2
o;
i
-6
H/IKI
(a)
Fig. 3.48
HIJ
0
(b)
Phase diagram of the AF triangular lattice in a magnetic field. (a) RS-RG and MC results ; (b) MHZ and MC results (refs. 88, 94,95).
Jnn
o:
.: I I I I ! I I J
2
L
I
Fig. 3.49
The
MC
3
0
,
,
,
,
=., J,n
Triangular lattice in a magnetic field for J2/Jl = - i . At IHI = H t tricritical points (crosses) occur where the order of the transition changes from second (below) to first (above) (ref. 88).
results
transition q =
,
I
-2
-~
for
Potts
Landau
also
and
at
the
but
this
we
are 0
< H
thus
consistent
< Ht
with
belongs
to
the
the
prediction
universality
that class
the of
phase the
model.
examined crossover do
not
the
critical
from discuss
XY
to
here.
exponents q
=
3
at Potts
the
tricritical
behavior
for
point small
H ,
96
A good
experimental
gular
lattice
which
Doukour~
scattering.
with
At
finally
for
sitions
, for
H > Hc2
interactions
Hcl
(see Fig.
of the
they
I × 2
monotonous
expected
3.47). and
and J2
the
I × 2
= 20 kOe (Fig.
for
T = O
trian-
ErGa 2
, for
and n e u t r o n
phase 2 x 2
3.50).
This
for phase
and
is e x a c t l y
in the r a n g e (above
the
the m a g n e t i s a t i o n
of
(Fig.
ErG2o/
is
the
temperatures
phases)
saturation
s y s t e m on the
magnetisation
phase
At h i g h e r
2 × 2
until
Ising
J1
find
< H < Hc2
the f e r r o / p a r a
of t r a n s i t i o n s
I/4 < ~ < I
increases
negative
low t e m p e r a t u r e
= 6.8 kOe
sequence
for a f r u s t r a t e d
and G i g n o u x 80 h a v e m e a s u r e d
H < Hcl
the
example
tranErGa 2
3.50).
K'~
N
0
Fig. 3.50
A more
precise no p h a s e
boundary
test
between
the
approximation as has b e e n
in the
chapter
itatively
for this
diagram
3.47 N a k a n i s h i
field tems
8 12 t6 20 2& APPLIED FIELD (kOe)
28
Magnetisation M of ErGa 2 as a function of the magnetic field (ref. 80). For low temperatures two critical magnetic fields occur where M rises abruptly.
H % 0
Fig.
&
been
2 x 2
and
on
is not
which,
3d
3 x 3
however,
where
For
(or
yields
this
because
the v i c i n i t y
I × 3) p h a s e s
modulated
for the A N N N I - c a s e . systems
yet p o s s i b l e ,
determined.
and S h i b a 96 d i s c u s s
proven
correct.
system
has
wrong
phases results
We come
back
approximation
of
for the
in
within for
mean
2d
sys-
to this
paper
should
be q u a l
97
~2K~2£H2~i~
3.4.3c A
2d
Ising
configuration represent tion
of a s u b m o n o l a y e r
causes
ic field becomes age
8
Gas M o d e l
spin c o n f i g u r a t i o n
occupied
J1
Lattice
sites,
of adatoms
si = - I
repulsion
between
the c h e m i c a l
(~ m a g n e t i s a t i o n
can be taken
M).
as a r e p r e s e n t a t i o n
on a surface:
vacant
ones.
two adatoms
potential
The
spins
AF
nn-interac-
on nn-sites,
determining
Such an a d s o r p t i o n
the magnet-
the average
model
of a si = + I
cover-
is called
a lat-
tice gas model. For a m o r e Kr
realistic
on h e x a g o n a l
nnn-interactions
description
graphite
of the a d s o r p t i o n
layers
(Fig.
as an a p p r o x i m a t i o n
3.51)
of noble
one needs
for the better
gases
like
at least also
Lennard-Jones
po-
tentials 97 .
Fig. 3.51 Lattice gas model with nn- and nnn-interact±ons for the description of adsorption of noble gas atoms on hexagonal graphite layers (ref. 97).
It m a y well be p o s s i b l e necessary
refer
d i agram s
lead to v e r y
ranges
the m a p p i n g
preted;
further
reaching
of a d s o r p t i o n
complicated
phase
interactions
~th
first
of the p a r a m e t e r s
on the
one only has
lattice
lattice
to note
gas
found
are
layer m e a s u r e m e n t s ,
diagrams.
Therefore,
to the p a p e r by K a n a m o r i 98 who has d e t e r m i n e d
for the h e x a g o n
and in special havior 99 . After
still
for the i n t e r p r e t a t i o n
these w o u l d here
that
GS
but
we
phase
to third n n - i n t e r a c t i o n s "devil's
the Ising results
that e x p e r i m e n t a l l y
staircase"
be-
can be r e i n t e r -
in a d s o r p t i o n
layers
98
the
coverage
8
(~ M)
independent
variable.
(Fig.
this
3.52)
and
not
For
leads
the
chemical
i = - I
to
large
.°. ~
in
potential
the
8
coexistence
- T
u
(~ H)
phase
is 88
diagram
the
regions.
LL
_]~9_o. • (13
0.5
e
~co~
J- :
> IJnnl) (ref. 88).
washed
and
is
3.53
covering
become
Langmuir
surfaces
si = +
successive
steps
the
theories
by
in F i g .
approach
on
binary
the
these
A review
Further
shown
Fig. 3.53
in
results and
B
are
limit
rapid
can
adatoms.
layers
can
on
solid
development be
be
usually
T < T 2,
applied
are
represented 3d
systems,
99
~{2_~b!£S_~!~h_~2~£~!~_~z_~_~:!~2~£~2£!2~£z_~2!~:
3.4.4
tion .
In this arise
.
.
.
to V e r t e x .
.
.
.
.
section
.
.
.
.
.
Models .
but
.
.
F irst
Dalton
lattices
with
J1 > O
occur,
model.
on the
8-vertex
is b r i e f l y
discussed,
generacies
of the
additi o n
we refer
who studied actions model,
as well
especially
expansions
the critical
respective
2d
surfaces
points
between
of a
exponents emerge.
16-vertex
6-vertex
are
this has
critical
the
for
the
models GS
de-
model. In of M i y a s h i t a I02 and Fujiki et al. I03
of a d d i t i o n a l
regard
odd
exponents
. However,
field t r i c r i t i c a l
and the
frustrated
with
series
as the c o n n e c t i o n
to two papers
the effect
from
one n o n u n i v e r s a l
(Baxter)
latter m o d e l
in the fully
in V i l l a i n ' s
not.
I = J2/J1
magnetic
does not
nn- and n n n - i n t e r a c t i o n s
for one of the two c r i t i c a l
On the other
and in an a d d i t i o n a l
The m a p p i n g
or
, that
of the ratio only
frustration
as for example
concluded
J2 > 0
where
between
field p r e s e n t
and
independent
general
Ising m o d e l s
and Wood I01
turned out to be true more
.
by c o m p e t i t i o n
in a m a g n e t i c
completely
.
nn-interactions
is caused
J1 and J2 e.g.
.
we consider
from c o m p e t i n g
model,
.
third
nn AF
(ice)
and fourth
triangular
to the o c c u r e n c e
nn-inter-
and V i l l a i n ' s
of an
XY
odd
like phase
transition.
3.4.4a The
S[s~2m_W!~h2~_Ma@netic
GS
phase
diagram
and n n n - i n t e r a c t i o n s corresponding Fig.
of the square
shows
to the
Field
three
I x I , ~2 × ~2
3.54 the t r a n s i t i o n
lines
Because
the order p a r a m e t e r
and M u k a m e l 7 have p r e d i c t e d to the u n i v e r s a l i t y
of the
class
renormalization
space
block
spins g e n e r a t e
fore,
the m o d e l
(J1' J2'
additional
has been
J4 ) . This
space
studied
should
group
formulas
structures IO4
with
transition
SAF) In
cubic
(RS-RG)
anisotropy
m e t h o d s IO4-IO6
interactions
in the e n l a r g e d
to
occur.
for the i n t e r a c t i o n s
four-spin
also c o n t a i n s
nn-
(F, AF,
is t w o - d i m e n s i o n a l ,
of the X Y - m o d e l
When
real
I × 2
phase
exponents
out that the r e c u r s i o n
system with ranges
the c o r r e s p o n d i n g
critical
using
Ising
schematically.
SAF
and thus n o n u n i v e r s a l
it turns
and
are shown
Krinsky belong
lattice
low t e m p e r a t u r e
J4
parameter
the special
case
between There-
space J1 = 0
100 K2
AF
"~
s..__Z
YT
~YT
KI
SA~
SAF Fig. 3.54
which
Kadanoff
8-vertex J2'
Phase diagram of the square lattice with nn- and nnn-interactions (ref. 104).
model
J4 ~ O
a n d W e g n e r I07 h a v e solved
The B a x t e r m o d e l tical
exponents
dicating 3.55
a whole
shows
exactly
is a l s o c a l l e d has
line
the t w o
of
sheets
t o be e q u i v a l e n t
b y B a x t e r I08.
Baxter
a second
depending
proven
the
Ising model
with
nonuniversal
cri-
model.
order
transition
continuously fixpoints of t h e
Thus
to the
like
with
on the r a t i o : in the
critical
2d
surface
~ = J4/J2 XY-model.
in
, inFigure
( K I , K 2 , K 4)
space.
K2 Kq
Fig. 3.55
The two sheets of the critical surface of the square lattice with nnn and 4-spin interactions (ref. 105).
nn ,
101
Figure
3.54
sheets
there
upper J1
corresponds
of
these
fixpoints
one
finds
nonuniversal
on the
the r e a s o n
lower
why
J1
# 0
As
an e x a m p l e
sheet
specific
heat
tubation
theory
3.56
is s h o w n
nn and
therefore,
behavior
thus
On b o t h
case).
only
for
of f i x p o i n t s
sheet,
3.55.
(Baxter
In the
exactly
J4 # 0
is a t t r a c t i v e .
also
for
for
. Contrary This
J4 = 0
is
and
is f o u n d I06
the d e p e n d e n c e
as f u n c t i o n
by B a r b e r I09.
independent
garithmically,
line
lower
in Fig.
KI = O
critical
the
behavior
in Fig.
at
are r e p u l s i v e ,
on the w h o l e
nonuniversal
K4 = 0
fixpoints
sheet
= 0
to this
two
to the p l a n e
is a line
For
Ising
square
s(J1=O)
= 0
of J1
of the e x p o n e n t
J1/rJ21 = O
of the
, as d e t e r m i n e d
the
lattices.
~
system
Then
c
in p e r -
decomposes diverges
into
only
lo-
.
01.
C~ (}3 02 01
o~
Fig. 3.56
For
d:3
d~.
ds
Nonuniversal variation of the exponent i = Jl/IJ21 (ref. 109).
finite
ate
' a'z
magnetic
2 x 2
phase
(Fig.
3.57).
order
in e v e r y
The
field
emerges GS
chain
field)
just
chains
are not
correlated,
chains
are
ordered
F
and
of this
second
as the
H
between
SAF
2 x 2
(with
phase.
spins
of the specific heat with
an a d d i t i o n a l
F, AF and phase
spins
in the
the
Id
degener-
SAF
phases
exhibits
perfect
parallel
However,
whereas
(with
T = O the
e
intermediate
SAF
antiparallel
ferro
to the m a g n e t i c
phase to the
the
AF
ordered
intermediate
field).
102
Fi 9. 3.57
The is
GS
We
=
diagram
to
the of
diagram
now
(J1
In
phase
degeneracy
phase
to
consider , where and
~ >
first
case
shown
field
in F i g .
result
the
the
I/2
critical
(Fig.
3.57)
iota. ~ ,_%'.-,,.
× 2
phase
the
phase.
Hc2 3.58
have
(l < O) =
4
(ref.
from
11Oa,
a correction
of
who
did
not
Brandt 11Ob
has
extended
yet
scale
discuss this
GS
neighbors.
transitions
cases
I/2
apart
of K a n a m o r i
2
third-nearest
< O)
the
,AL
-5_/_~ T
GS phase diagram of the square lattice with nn- and nnn-interactions Jl and J2 in a magnetic field H .
identical
the
,
-~
-'
for
I = J2/J1 to the
IJ11 111).
. . . . . . . . .
be
AF
< O
nn-interactions
, ~ = O
, 0
< ~ < I/2
,
distinguished. GS
changes
from
AF
. The
phase
diagram
for
For
small
H/J I
there
to
at
the
I = - I
is
is
F
an
Ising
Tncr~ti ca! I~nt ~ramagn~tt
IOrderecl lanfifmromgnetl i I/ r
Fig. 3.58
Phase diagram of the square lattice with (ref. iii).
I = - i
in a magnetic
field
103
transition
with
normal
field
with
modified
Ht
first
order
(dashed
been
treated
For
I > O
with
on t h e
lieved with
interesting
(that is
of M ~ l l e r - H a r t m a n n
calculation
also
of d o m a i n
We mentioned lattice
series
Above
the
the
tricritical
transition
tricritical
becomes
behavior
has
of a u t h o r s 112
I = O
to b e exact.
3.59)
The
line).
exponents
a n d L a n d a u 113 h a v e
the triangular
(Fig. are
line).
a n d for
the r e s u l t s
based
(full
critical
by a number Binder
culations,
exponents
carried J2 = O)
out
and
Zittartz
wall
energies
this method
MC
a n d at f i r s t w e r e
be-
in c o n n e c t i o n
In the p h a s e
r e s u l t s 114 a n d
cal-
excellent agreement 76 (MHZ) which are
already
and the ANNNI-model.
expansion
extensive
found
RS-RG
diagram
r e s u l t s 115
included.
10
20
kBT/ IJnn~
In t h e m e a n t i m e determined square
z
=
Contrary Fig.
=
116
the
e
by high
critical
order
activity
series
expansion
zc
of t h e h a r d
± O.0001
have
the
MHZ
(3.92)
result
= d(H/J1)/d(T/J I)
a value
MHZ
zc
precisely
to t h i s m
et al.
gas:
3.7962
3.59,
yields
Baxter
very
lattice
c
square lattice in a magnetic field. Circles method; dashed line: series expansion;
Phase diagram of the nn AF MC results; full line: MHZ points: RS-RG (ref. 113).
Fig. 3.59
for -2m
the
=
critical
4
for t h e
slope
at t h e p o i n t
of the
(H = 4
curve
in
IJ1 I, T = O)
activity
(3.93)
104
Therefore,
the
analytical
approximation.
In
the
range
at
two
critical
MHZ
0
method
< I < I/2 fields
p h a s e as c a n b e s e e n d a u 113 f o r i = I/4 and
H e m m e r 117
show in
that
ly w r o n g
T > 0
the
SAF
T = O).
and
Fig.
(Fig.
be
it
increasing
field
H
Hc4
AF
from
3.57.
3.60a)
exact,
MC
and
agree
quite
line
Hc3
field
approximation
the
results
the
3.61)
MHZ
well
P
to
good
the
GS
changes
to
F = P
of
Binder
the
of AF
reaches
which
a very
2 x 2
results
for
phase
is
and
phase
down
yields
Lan-
Doczi-Reger and
to
T = 0
a topological-
diagram.
there
is
phases
no
as
Therefore,
sharp
both
at
order-disorder
in F i g s .
but
with
Hc3
from
the
to m e a n
phase
For
an
(Fig.
along
contrast
cannot
transition
have
T = 0
the
same
between
transition
occurs
between
the
2 x 2
symmetry
(as
opposed
Hc3 shown
respective as
thick
line
and to
Hc5
and
at
T = O
Hc4
3.6Oa,b,c.
Ca) H
ciegenerote tructure
I = 1/4
(b) i.~..i~
5.(]
X = 1/2
dege~ote
UNN'
structure
H
C
2.5
5.0 t ~ ' ~ .
OC
(=)
H
I
2
degenerate
t
IJ.NI structure 10.0 ~ /
00 .
kBTI,JA,
=
0
I =
I/2
paramagnetic (T = O) sition
and to
the
,
~
(Fig.
SAF P
kBTl,J ~,
3.60b)
to
.
.
.
1
ksTIIJ~l
Phase diagram of the square lattice with nn- and nnn-interactions in a magnetic field for (a) : I = I/4 ; (b) : I = i/2 ; (c) : I = i . All phase transitions are first order (ref. 113).
r
down
.
05
Fig. 3.60
5.0 2.5
For
.
P
1
7.5
00
.
"
and
T = O
phase phase.
H
< Hc3
, between
(T > O)
= Hc5 Hc3
occurs,
and with
the
system
Hc4
the
a second
remains 2 x 2 order
phase tran-
105
70 O8 06 OL O2 0
Fig. 3.61
For
I > I/2 SAF
second
order
(Fig.
agreement
size
with
In the
limits the
H = 0
and
•
the
and
in l o w f i e l d s
H > O
one contiguous
and Landau
transition,
for
have
whereas
find nonuniversal
~ = i/4 .
the
system
transition ~ < I/2
found
.
normal
for the
critical
is
line of
2d
SAF-P
exponents
in
results.
and
(for f i x e d
l)
H ~ ~
the exponents
ap-
values.
to V e r t e x
to n o t e
field
tensively
I Z H'
only
Binder
they
I/~ ~ 0
Ising
Here we want
field
I I
to t h e b e h a v i o r
AF-P
H @ O
Connection
without
contrary
analysis
the
already
T > O
at the
for
proach
3.4.4c
for
occurs,
exponents
transition
3.6Oc)
phase;
finite
Ising
I "1
The AF phase transition in MHZ approximation (line) for The points are the MC results from Fig. 3.60 (ref. 117).
in t h e
From
! "2
the
the
close
8-vertex
16-vertex
and w h i c h
Models
model,
connection model
and
which
Lieb
we mentioned
between
the
that between
Ising the
a n d W u 118 h a v e
already
when
of t h e
Ising
system
system with described
discussing
ex-
the Baxter
model. Lieb tions
and Wu J1
show the
a n d J2
equivalence
' the n n n - i n t e r a c t i o n s
system with
J a n d J'
, the
nn-interac-
four-spin
106
interaction the
first
J
and
eight
responding
a constant
vertices
to the v e r t i c e s
interactions
J
of Fig. are
O
, with
the
3.62 m a y
linear
8-vertex
occur.
reversible
The
model
in w h i c h
energies
e. c o r l of the I s i n g
functions
J. 3
+ + + + + + + + .....- -
{91
00)
(ll)
.....
(ia)
(1~)
~ ....... F
(:4)
05)
06)
+ + + + + + + + + u .....~...... , a, ,,+ + + Fig. 3.62
Lieb
The sixteen vertex configurations of the general ferroelectric model on the square lattice and the corresponding bond configurations using vertex (i) as basis (ref. 118).
and Wu also
interactions special Fig.
of the
16-vertex
occur.
These
KDP
six v e r t i c e s corresponds
of Fig.
3.62
lattice
of
T = O
fully
in T a b l e
square,
where
cases
all
3.2f,
where
nnn-
to be e q u i v a l e n t
to t w o
sixteen
of
are called
Ising
which
determined
for
T = O
requires
vertices
the g e n e r a l i z e d
system This
. The
system
two
ice rule. 120 exactly :
GS
because
3.2f
four
F
away
all pair
spins
first cases
six v e r t i c e s
arrows
at e a c h
from
it 121
interactions
c a n be i n t e r p r e t e d tetrahedra
the
in t h e s e these
t w o to p o i n t
cornersharing
tetrahedron to the
of T a b l e
such that only
t w o of the
it a n d the o t h e r
frustrated
in e a c h
c a n be c h o s e n
ice m o d e l 1 1 9 ' 1 2 0 ,
strength.
just corresponding Lieb has
occur
square
towards
In t h e e q u i v a l e n t a n d of e q u a l
model
special
the p a r a m e t e r s
to the
to p o i n t
shown
other
model.
o b e y the ' i c e - r u l e ' ,
vertex
For
Ising model
in e v e r y
cases
For both models
AF
only
3.62 m a y
respective
must
show the
occur
as a
(see Fig.
are
2d 3.63).
are u p a n d t w o a r e down,
The entropy
of t h e
square
ice m o d e l
107
Fig. 3.63
whereas single
3
4 in
the
~4
~
simple
tetrahedra
NP g
I I
I
I
The square lattice (bottom) is equivalent to the sharing tetrahedra (top).
_
So
i I
O.2158
Pauling
2N {3~ N/2 Tc
. Both
to the v a n i s h i n g
exactly
shall
a n d Wu.
properties
ice r u l e 118
effects
and three-spin
order
diverging
on the c r i t i c a l the
frustration
nn-pair
of i n f i n i t e
by Lieb model
like
lie e x a c t l y
general
thermodynamic
exponentially
decays
longer
with
interesting
discussed
G(r)
ice r u l e
dels which
the
order with
extensively
tion
Ising
(e.g.
of f i r s t
are
further
chapter.
b y B a x t e r a n d W u 122 a n d 123 and Hemmer u s i n g the
with
four-spin
interactions
109
4.
Three-Dimensional
In the
last chapter
frustrated three
Ising
Ising
hcp-lattice
on the
fcc
lattice,
a l m o s t no exact
three
, the simple
which
4.4 the
3d
from the
4.1
ANNNI-model 2d
of results
cubic
are known.
First
(sc)
with
for
2d
to this in
An e x c e p t i o n is the on the fcc-
in Section
systems
transitions follows,
Contrary
interactions
4.5.
frustrated
show phase
markedly
results
four-spin
in Section
fully
a multitude
m a n y of them are exact.
system with
discussed
4.3 we treat
Ising Systems
we have d e s c r i b e d
systems,
dimensions
self-dual
Frustrated
4.1
nn-pair
and
to Section
interactions
and the p y r o c h l o r e
(=B
of d i f f e r e n t
In Section
which
order.
has p r o p e r t i e s
spinell)
differing
case.
f_cc A n t i f e r r o m a g n e t
For this
system with
the
to be only
GS
can be stacked
AF 2d
nn-interactions ordered:
arbitrarily
J1
perfectly
on each
other.
D a n i e l i a n 124 has shown AF
ordered
This yields
100
a
GS
planes degener-
acy Ng
~
of course and high
2 (NI/3) the
GS
temperature
for
series
GS
The i n t e r e s t i n g ly has been
Different
question
diverging
series,
for
number
vanishes
(~ N-a/3)
Danielian
gave a rough e s t i m a t e
rJ1r
. Betts
. From
averaged
(1.83±O.O2)
over
The spin GS
how to do the low t e m p e r a t u r e e x p a n s i o n correctauthors 126-128.
for the e x i s t e n c e
N ~ ~
must
differ
They have
formulat
of such an expansion:
from each other by a
of spins with d i f f e r e n t
c o n f i g u r a t i o n s of low e n e r g y
m a y differ
a small
from the c o n f i g u r a t i o n
(finite)
all
fJ1 r
orientation. (4.2)
(b)
low
and E l l i o t t 125
but u n c o r r e c t l y
Tc ~
by several
conditions GS
T c ~ 1.2
and o b t a i n e d
investigated
ed the f o l l o w i n g
N ~ ~
expansions
temperature:
the low t e m p e r a t u r e
(nonequivalent)
(a)
(4.1)
entropy
of the t r a n s i t i o n extended
,
number
excitations of this
GS
'near'
one
for only
of sites. (4.3)
110
When
these
fa(T)
conditions
starting
the e x c i t a t i o n Mackenzie that
for
phases All
fulfilled
GS
small
J2
127
(a)
have
only
with maximum phases
can o n l y
are a
two
(with
Id
that
these metastable appear
equally
several whereas
2d for
of the
free
sality
class
This
phase
the
free
depends
energy
on
s
via
and
showed
2d
transition
sc-sublattices
have
spin
disorder.
between
phase
model
the o r d e r e d
up and
the o t h e r
but
n
two
so
cubic
spin
systems
d = 3
and
in the u n i v e r -
anisotropy.
where spins
tem-
the L a n d a u
. For
transition
phase
finite
and
frustrated
with
and
are
at low t e m p e r a -
properties
dimension
2d
Heisenberg
spins
at low,
Id GS
energy
phases.
occurs
the
an e f f e c t i v e l y
of the
simulations
for g e n e r a l i z e d and
free
differences
equilibrium
is
degenerate
thermodynamically.
a higher energy
MC
LRO
there
fcc-lattices
expect
the
J2
twelfefold
stable
have
in
studied
energy
are
as the
systems
expansion
on d - d i m e n s i o n a l
phases
T = O
and P i n c u s 8 h a v e
they
fa(T)
respective LRO)
However,
stable
Alexander
n = I
(and
disorder)
will as for
can d e t e r m i n e
a nnn-interaction
sixfold
be m e t a s t a b l e .
tures
peratures,
included
symmetry
small
Just
one
; in g e n e r a l
spectrum.
and Y o u n g
other
thus
from
two of the down,
four
and the
P
p h a s e has b e e n i n v e s t i g a t e d in a series of MC p a p e r s by P h a n i et 129 130-132 al. and B i n d e r (et al.) , w h o also i n c l u d e d a n n n - i n t e r a c t i o n J2
and
For
a magnetic
H = J2 = 0
T/IJ1l
= 1.75
critical At
these
remains
For Hcl
a first
. With
fields
Hcl
critical
from
~
< H < Hc2
transitions del 133 "
the
the
=
transition
field and
GS
to
H
Hc2
the = 12
entropy
T = 0
occurs GS
at
changes
IJ1i
is f i n i t e 132 and
the
system
(4 4)
transition
is t h r e e f o l d
Corresponding
be as in the
4.1).
:
the p h a s e phase
at two
(see Fig.
~I in 2
ordered
fourfold.
should
phase
IJiL
two p o i n t s
0 < H < Hcl
.
order
down
So(Hc2)
these
H
increasing = 4
fields
paramagnetic
So(Hcl) Apart
field
q = 3
to t h e s e
is a l w a y s
first
degenerate, degeneracies,
respective
q = 4
order.
for the p h a s e
Potts
mo-
111
H¢2113.1 10 HIl.l"..I 5 1 Fig. 4.1
Domany
MC phase diagram of the (ref. 130).
e t al.
133
a nnn-interaction come
together
this
behavior
tions, Fig.
had
2 keT/Lf..I i
AF
predicted
J2
(> O)
fcc-model in a magnetic field
a new kind
is added.
at a m u l t i c r i t i c a l p o i n t 132 Binder has f o u n d for
although
without
determination
H
of c r i t i c a l
behavior
two phase
transition
The
(Tm > O,
H = O)
J2 = - J1 of the
in
critical
MC
when line~
Exactly calcula-
exponents
(see
4.2).
H¢2 10 H
R=-I ,J
I]...,I
"c_,_S
I]n.I
ill/
,
,
The
fcc
describe are
MC phase diagram of the J2 (= - Jl ) (ref. 132).
Ising AxBI_ x
contained
system with alloys.
in the
MC
,
]
_
& kmTiiJn. I 6
2
Fig. 4.2
,
AF
J1
II
fcc-model with additional nnn-interaction~
< O
and
J2
> O
Correspondig
results
papers
here.
cited
and
has
been
further
used
to
reference~
112
4.2
Fully
and P a r t i a l l y F r u s t r a t e d
The p r o p e r t i e s (sc) As
Ising
square
of the
system
along
are
of
frustration
of e i t h e r
ented
F and AF
contains
for
to look
4.3
respective
frustrated
such
simple
for s i m p l e
or
cubic
as in the f c c - c a s e .
an odd n u m b e r
on the
all p l a q u e t t e s ,
In Fig.
four
only
one has
Lattice
established
nn-interactions
(a)
in the X Y - p l a n e . cell
as w e l l
frustrated
the edges,
figurations
Cubic
and the p a r t i a l l y
are not y e t
plaquettes
interactions
unit
fully
Simple
of
sc-lattice (b)
only
configurations
two e l e m e n t a r y
AF
periodic
con-
leading
those
are
to
ori-
shown.
The
cubes.
(a) "
Yt, Fig. 4.3
et al.
frustrated because
X
134
have
system
one
just
GS
diamond
other
SO
to one
GS
~
from
N-I/s
the
they
for
must
d > 3 or
effect
d i s o r d e r 136 : T h e r e parallel
reasons
interaction
degeneracy
frustrated
determined
for d - d i m e n s i o n a l
as for
of f r u s t r a t i o n The
)
of t o p o l o g i c a l
than
state,
(c)
(a) Comb representation of the fully frustrated square lattice. Straight (wavy) lines represent F (AF) interactions. Elementary cubes of the partially (b) and the fully (c) frustrated sc-lattice are obtained by stacking the square lattice of (a) in different ways (ref. 135).
Derrida
more
(b)
be
4
have
is h i g h e r lattice58).
1OO
the p r e v i o u s
(sc)
hypercubic d > 4
lattices
in p a r t
and
'superblocking'.
GS
of
where
strong
AF
fcc-lattice
Id
disorder
one
every
fourth
chain
can be f l i p p e d leading
found
that
unfavourable
called
in the
fully
of the p l a q u e t t e s
f c c - l a t t i c e 8. This
one,
,
of the
in the e n e r g e t i c a l l y
Instead
direction
properties
in the
then
is a set of
GS
as a w h o l e
kind
(and in the finds
2d
of spins
to get an-
to
(4.5a)
113
in c o n t r a s t
S
~
o
to
N-2/s
(fcc,
This
has an i m p o r t a n t
When
only a finite
one
GS
bE
that
=
4
of these L
MC
of length
L
additional
energy
low e x c i t e d
states.
of such a chain arises
is flipped
a second
from
only at the ends
of
(4.6)
'one-dimensional'
expansion
in good
for the
IJJ
calculations
cate
part
(4.5b)
is the d i f f e r e n c e
. But this way
series
consequence
configuration,
this part;
diamond)
excitations
condition
(4.3)
is i n d e p e n d e n t
is violated,
of the
length
and no low t e m p e r a t u r e
exists. of B h a n o t
order
and Creutz 137 and of K i r k p a t r i c k 136 indi-
transition,
but the c r i t i c a l
temperatures
are not
agreement:
Tc/IJl
~
0.8
(ref.
137)
, (4.7)
Tc/IJl
~
1.25
(ref.
w h e r e a d d i t i o n a l data are in favour Chui et al. mation tain
138
using
a much
This ing
~
that
to d e t e r m i n e but
and specific
Tc
from a mean
(contrary
(ref.
heat
to their
(Fig.
4.4)
field
approxi-
claim)
they ob-
systems
138)
with
are not well
and a p a r t i a l l y
(4.8)
strong
frustration
described
et al. 135 have d e t e r m i n e d
the f o l l o w i n g
Fully
factor
value
fluctuations
for the fully predict
tried
,
value.
sublattices,
2.4
demonstrates large
Blankschtein
(a)
have
higher
TMF, c /IJi
for s t r u c t u r e
of the higher
eight
136)
by m e a n
and c o r r e s p o n d field
the G i n z b u r g - W i l s o n
frustrated
sc-system
theory. Hamiltonian
(see Fig.
4.3)
and
behavior:
frustrated:
The order
parameter
expansion
up to
~2
has
four c o m p o n e n t s
is c o n s i s t e n t
with
(n = 4)
; ~ (5 4-d)-RG
a transition
of
(weak)
114
1.2
1.0
2
: ~ I/3
into three
sublattice
are o r d e r e d In 2)
low temperature
J2
. The properties
the
2d
case discussed
vestigated survey
very
temperature
from
series
MC
(in
of t h i s
The
basic
ferromagnetic
to
The point
4.14.
. Chui
phase,
but
are
ferromagassumes this
re-
z
The nn-interactions
direction)
interact
3d
in S e c t i o n 149-151
system
3.4.1.
phase
occurs
For at
(T = O,
a multiphase
antiferromagnetically
This
diagram
T = 0 ~
point with
Id
system
different has b e e n
can g i v e is s h o w n
high -154
a change
(= - J2/J1)
K = I/2)
J
o ferromagnet-
are
are quite
, and here we
calculations 152'153,
expansions.
d = I and 2 precisely
J1
extensively
of t h e r e s u l t s .
It is d e r i v e d
and
in Fig.
also nnn-spins
via
more
MC
PC
to N i 2+
to be p r o v e n .
4.4
ic;
on t h e
interest
as the
a cubic
T = 0
(e.g.
systems.
in
sc-sublattices.
one has
ions
T = 0
studied
forming
Cu
data
of m a g n e t i c
randomly
as o u r
predicting
frustrated
experimental
kinds
less
further,
paramagnetic
A related
or
this
remains
ture
more
discuss
PC
of t h e
of t h e t w o d i f f e r e n t
only
from
ina brief
in Fig. 4.15. l o w _155
and
of t h e
= I/2
, j u s t as
is a f r u s t r a t i o n degeneracy.
GS
from for
point,
or
123
Fig. 4 . 1 4
Unit cell of the
3d
ANNNI-model.
I
~;IJ,
!
p,RA
L
*'~z MODULATE0
z
"~ / / FERRO
'
Fig. 4.15
Fisher
Selke
nite
sequence
phases
156
have
(Fig.
4.16)
to t h e
and only
one modulated
Figs.
wavevector The
steps
'
o'.,,
3d
3.38
2d
which
! I
T
- 3.40).
that
in Fig.
,:_j~/j,
near
&
! 1.0
i
the
phase
with
the multiphase
occurs
all extend P
between
down phase
to
is s h o w n
the
T = O
extends
continuous
The discontinuous
q at l o w t e m p e r a t u r e s
its o r i g i n a l
'
phases
case where M
o'.,
ANNNI-model (ref. 151).
found
of c o m m e n s u r a t e
contrast
(see.
II
Phase diagram of the
and
12,21 ANTIPHASE
t /
o',2
z
point F
. This
down
to
wavevector
variation in Fig.
an i n f i
and
of t h e
the is in T = 0
exists average
4.17.
4.17 d o n o t f o r m a c o m p l e t e " d e v i l ' s s t a i r c a s e " 157 because not all rational numbers q/q
meaning
in
124 (zh)=I2,z,~)=~.,.ttlltltlllllll... Ic2(TJ
xI(T)~ (3,3) Antiphosl
12
3)
x3(T ) (2~3)
(2,21 Antipho~e
I{ • -Jz/Jt
Fig. 4.16
Schematic phase diagram of the point (ref. 156).
3d
ANNNI-model near the multiphase
I 10
:V-
q
q. q
o9
